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Digitized  by  the  Internet  Archive 

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TECHNICAL    DRAWING    SERIES 


DESCRIPTIVE    GEOMETRY 


BY 

GARDNER   C.    ANTHONY,   Sc.D. 

II 

GEORGE   F.    ASHLEY 


OF   T-HE 

UNIVERSITY 


sEdi-IFORNil 


BOSTON,  U.S.A. 

D.    C.    HEATH   &   CO^   PUBLISHERS 

1910 


GENERAL 


Copyright,  1909, 
By  D.  C.  Heath  &  Co. 


PREFACE 


An  extended  experience  in  engineering 
practice  and  teaching  has  emphasized  the 
importance  of  certain  methods  for  the  presen- 
tation of  principles  and  problems  in  graphics 
which  the  authors  have  gradually'  developed 
into  their  present  form.  Most  of  the  subject- 
matter  which  we  now  publish  has  been  pre- 
sented in  the  form  of  notes  and  used  during 
the  past  eight  years,  and  nearly  all  of  the 
problems  have  received  the  critical  test  of  the 
classroom.  Previous  to  the  preparation  of 
the  notes,  which  were  made  the  foundation 
of  this  treatise,  the  third  angle  of  projection 
had  been  adopted  for  problems,  and  almost 
exclusively  used,  as  conforming  more  nearly 
to  engineering  practice. 

It  has  been  the  aim  of  the  authors  to  make 
a  clear  and  concise  statement  of  the  princi- 
ples involved,  together  with  a  brief  analysis 
and  enumeration  of  the  steps  to  be  taken,  so 


that  the  essentials  of  each  problem  shall  be 
clearly  set  forth.  The  illustrations  have  been 
chosen  with  care  and  ari-anged  so  as  to  appear 
opposite  the  descriptive  text. 

Too  much  stress  cannot  be  laid  on  the  im- 
portance of  preparing  problems  with  such  care 
that  they  may  clearly  illustrate  the  principles 
involved.  In  general  it  should  not  be  neces- 
sary for  the  student  to  prescribe  the  condi- 
tions governing  the  relations  of  points,  lines, 
and  surfaces  because  of  the  time  consumed 
and  the  probable  failure  to  bring  out  the 
salient  features  of  the  problem.  The  graphic 
presentation  of  the  problems  to  be  solved 
should  facilitate  the  lay-out  by  the  student, 
and  enable. the  instructor  to  judge  quickly  of 
their  character  and  adaptability  to  the  special 
cases  under  consideration.  Two  sets  of  prob- 
lems have  been  prepared  to  illustrate  most 
cases,  and   the   number   may   be   further  in- 


196407 


IV 


PREFACE 


creased  by  reversing,  or  inverting,  the  illus- 
trations given.  Tlie  unit  of  measurement 
may  also  be  changed  to  adapt  the  problems 
to  any  chosen  size  of  plate,  and  if  the  propor- 
tion 1)3  maintained  it  will  be  possible  to  solve 
the  problems  within  the  given  space.  If  it  is 
desired  to  change  the  assignment  annuall}-, 
the  even  numbers  may  be  used  for  one  year 
and  the  odd  numbers  for  the  following  year. 
It  is  hoped  that  the  number  of  problems  is 
sufficiently  great  to  admit  of  considerable  va- 
riety in  the  course. 


Tufts  College,  January,  190*.). 


Although  the  authors  believe  that  an  ele- 
mentary course  in  tlie  orthographic  and  iso- 
metric projections  of  simple  objects  should 
precede  the  more  analytical  consideration  of 
line  and  plane,  as  herein  presented,  yet  this 
treatise  has  been  proven  adequate  to  meet  the 
demand  of  those  who  have  not  received  such 
preparation.  By  thus  making  it  available  for 
students  of  science  and  mathematics,  as  well 
as  for  engineers,  it  is  hoped  to  promote  a  more 
general  knowledge  of  the  grammatical  con- 
struction of  Graphic  Language. 

GARDNER  C.    ANTHONY 
GEORGE   F.   ASHLEY. 


TABLE   OF   CONTENTS 


ART. 
1. 

2. 
3. 
4. 


9. 
10. 


CHAPTER  I 
Definitions  and  First  Principles 


Descriptive  Greometry 

Projection 

Coordiuate  Planes    . 

Quadrants  or  Angles 

Orthographic  Projectioi 

Notation  . 

Points 

Lines 

Line  parallel  to  a  coordinate  plane 

Line  perpendicular  to  a  coordinate  plane 


1  n.  Line  lying  in  a  coordinate  plane 

1  ,  12.  Line  parallel  to  a  coordinate  plane 

2  13.  Lines  parallel  in  space 

3  14.  Lines  intersecting  in  space 

3  1.5.  Lines  intersecting  the  ground  line 

4  16.  Traces  of  a  line 

6  17.  To  define  the  position  of  a  line 

7  18.  Planes 

8  19.  GL  the  trace  of  V  and  H 
8 


8 
8 
8 
10 
10 
10 
10 
12 
12 


20. 

21. 
22. 

ii! 

24. 
2.5. 
26. 


CHAPTER   n 
Points,  Lines,  and  Planes 


Operations  required  for  the  solution  of  prob- 
lems       

To  determine  three  projections  of  a  line 

Projection  of  point  in  2  Q 

Revolving  of  P 

To  determine  the  traces  of  a  line     . 

To  determine  the  traces  of  a  line  parallel  to  P 

To  determine  the  projections  of  a  line  when 
its  traces  are  given 


14 
14 
15 
15 
16 
18 

18 


27.  Conditions  governing  lines  lying  in  a  plane    .       19 

28.  Conditions  governing  lines  lying  in  a  plane 

and  parallel  to  //or  F 

29.  An  infinite  number  of  planes  may  be  passed 

through  any  line  .         .         .      '  . 

30.  To  pa.ss  a  plane  through  two  intersecting  or 

parallel  lines.  Case  1 20 

31.  To  pass  a  plane  through  two  intersecting  or 

parallel  lines,  Case  2 20 


19 


20 


VI 


TABLE  OF   CONTENTS 


ART. 

32.  To  pass  a  plane  through  two  intersecting  or 

parallel  lines.  Case  3 

33.  To  pass  a  plane  through  a  line  and  a  point     . 

34.  To  pass  a  plane  through  three  points  not  in 

the  same  straight  line 

35.  Given  one  projection  of  a  line  lying  on  a  plane, 

to  determine  the  other  projection 

36.  Given  one  projection  of  a  point  lying  on  a 

plane,  to  determine  the  other  projection 

37.  To  locate  a  point  on  a  given  plane  at  a  given 

distance  from  the  coordinate  planes     . 
^38.    To  revolve  a  point  into  either  coordinate  plane 

39.  To  determine  the  true  length  of  a  line.  Case  1 

40.  To  determine  the  true  length  of  a  line,  Case  2 

41.  Relation  of  the  revolved  position  of  a  line  to 

its  trace 

42.  The  revolved  position  of  a  point  lying  in  a  plane 

43.  The  revolved  position  of  a  line  lying  in  a  plane 

44.  To  determine  the  angle  between  two  inter- 

secting lines ....... 

45.  To  draw  the  projections  of  any  polygon  . 

46.  Counter-revolution,  Construction  1  . 

47.  Counter-revolution,  Construction  2  . 

48.  Counter-revolution,  Construction  3  .• 

49.  To  determine  the  projections  of  the  line  of  in- 

tersection between  two  planes.     Principle  . 

50.  To  determine  the  projections  of  the  line  of  in- 

tersection between  two  planes.  Case  1  . 

51.  To  determine  the  projections  of  the  line  of  in- 

tersection between  two  planes.  Case  2 
•52.   To  determine  the  projections  of  the  line  of 
intersection  between  two  planes  when  two 
traces  are  parallel 


\(.K 

AUT. 

53. 

21 

22 

54. 

22 

55. 

22 

24 

.56. 

24 

57. 

25 

27 

58. 

28 

59. 

28 

30 

60. 

31 

61. 

31 

32 

62. 

32 

33 

34 

63. 

64. 

35 

65. 

35 

66. 

36 

67. 

68. 


37 


To  determine  the  projections  of  the  line  of  in- 
tersection between  two  planes  when  all 
traces  meet  in  (jL 37 

To  determine  the  projections  of  the  line  of  in- 
tersection between  two  planes.  Case  3  .       38 

To  determine  the  projections  of  the  line  of  in- 
tersection between  two  planes  when  one 
plane  contains  (iL 38 

Revolution,  Quadrants,  and  Counter-revolu- 
tion         40 

To  determine  the  point  in  which  a  line  pierces 
a  plane.     General  Method    ....       42 

To  determine  the  point  in  which  a  line  pierces 
a  plane.  Case  1 .42 

To  determine  the  point  in  wliicii  a  line  pierces 
a  plane.  Case  2 42 

To  determine  the  point  in  which  a  line  piel-ces 
a  plane.  Case  3      .         .         .         .         .         .42 

To  determine  the  point  in  which  a  line  pierces 
a  plane.  Case  4 44 

If  a  right  line  is  perpendicular  to  a  plane,  the 
projections  of  that  line  will  be  peipendicu- 
lar  to  the  traces  of  tlie  plane         ...       44 

To  project  a  point  on  to  an  oblique  plane        .       44 

To  project  a  given  line  on  to  a  given  oblique 
plane 45 

To  determine  the  shortest  distance  from  a 
point  to  a  plane    .         .         .         .         .         .45 

Shades  and  Shadows 46 

To  determine  the  shadow  of  a  point  on  a  given 
surface 47 

To  determine  the  shadow  of  a  line  upon  a 
given  surface 47 


TABLE  OF   CONTEXTS 


Vll 


ART.  PAGE 

69.   To  determine  the  shadow  of  a  solid  upon  a 

given  surface '48 

'       70.   Through  a  point  or  line  to  pass  a  plane  hav- 
ing a  defined  relation  to  a  given  line  or 

plane .50 

'  71.   To  pass  a  plane  through  a  given  point  parallel 

to  a  given  plane 50 

^  72.   To  pass  a  plane  through  a  given  point  perpen- 
dicular to  a  given  line 50 

73.  To  pass  a  plane  through  a  given  point  parallel 

to  two  given  lines 51 

74.  To  pass  a  plane  through  a  given  line  parallel 

to,  another  given  line 52 

-475.   To  pass  a  plane  through  a  given  line  pei-pen- 
dicular  to  a  given  plane        .... 

76.  Special  conditions  and  methods  of  Art.  70 

77.  To  pass  a  plane  through  a  given  point  perpen- 

dicular to  a  given  line.     Special  case  . 

78.  To  pass  a  plane  through  a  given  line  perpen- 

dicular to  a  given  plane.     Special  case 

79.  To  determine  the  projections  and  true  length 

of  the  line  measuring  the  shortest  distance 
between  two  right  lines  not  in  the  same 
plane     .        .        .        .        .        .        .        .      .54 

.—480.   To  determine  the  angle  between  a  line  and  a 

plane 54 


52 
52 
5.3 


81 .  To  determine  the  angle  between  a  line  and  the 

coordinate  planes 

82.  To  determine  the  projections  of  a  line  of  defi- 

nite length  passing  through  a  given  point 
and  making  given  angles  with  the  coordi- 
nate planes 56 

83.  To  determine  the  angle  between  two  planes. 

Principle 57 

84.  To  determine  the  angle  between  two  oblique 

planes 

85.  To  determine  the  angle  between  two  oblique 

planes  by  perpendiculars      .... 

86.  To  determine  the  angle  between  an  inclined 

plane  and  either  coordinate  plane 

87.  To  determine  the  bevels  for  the  correct  cuts, 

the  lengths  of  hip  and  jack  rafters,  and  the 
bevels  for  the  purlins  for  a  hip  roof     . 

88.  Given  one  trace  of  a  plane,  and  the  angle  be- 

tween the  plane  and  the  coordinate  plane, 
to  determine  the  other  trace.  Case  1    . 

89.  Given  one  trace  of  a  plane,  and  the  angle  be- 

tween the  plane  and  the  coordinate  plane, 
to  determine  the  other  trace,  Case  2    . 

90.  To  determine  the  traces  of  a  plane,  knowing 

the  angles  which  the  plane  makes  with  both 
coordinate  planes 

91.  Fourth  construction  for  counter-revolution 


55^ 


-„^ 


57 


58 


58 


60 


62 


62 


CHAPTER  ni 
Generation  and  Classification  of  Surfaces 


92.  Method  of  generating  surfaces 

93.  Classification  of  Surfaces 


65 
65 


94.  Ruled  Surfaces 

95.  Plane  Surfaces 


65 
66 


Vlll 


TABLE  OF  CONTENTS 


96.  Single-curved  Surfaces 

97.  Conical  Surfaces  . 

98.  Cylindrical  Surfaces 

99.  Convolute  Surfaces 


66  100.  A  Warped  Surface 

66  101.  Types  of  Warped  Surfaces 

67  102.  A  Surface  of  Revolution 
67  103.  Double-curved  Surfaces . 


PAGE 

68 
68 
70 
70 


CHAPTER  IV 
Tangent  Planes 


104.  Plane  tangent  to  a  single-curved  surface       .      72 

105.  One  projection  of  a  point  on  a  single-curved 

surface  being  given,  it  is  required  to  paas 
a  plane  tangent  to  the  surface  at  the  ele- 
ment containing  the  given  point        .         .       73 

106.  To  pass  a  plane  tangent  to  a  cone  and  through 

a  given  point  outside  its  surface         .         .       74 

107.  To  pass  a  plane  tangent  to  a  cone  and  parallel 

to  a  given  line     .         .         .         .         .         .75 

108.  To  pass  a  plane  tangent  to  a  cylinder  and 

through  a  given  point  outside  its  surface  .       75 

109.  To  pass  a  plane  tangent  to  a  cylinder  and 

parallel  to  a  given  line        ....      76 


110.  Plane  tangent  to  a  double-curved  surface      .      77 

111.  One  projection  of  a  point  on  the  surface  of  a 

double-curved  surface  of  revolution  being 
given,  it  is  required  to  pass  a  plane  tangent 
to  the  surface  at  that  point         ...       78 

112.  Through  a  point  in  space  to  pass  a  plane 

tangent  to  a  given  parallel  of  a  double- 
curved  surface  of  revolution        ...       79 

113.  To  pass  a  plane  tangent  to  a  sphere  at  a  given 

point  on  its  surface 80 

114.  Through  a  given  line  to  pass  planes  tangent 

to  a  sphere 80 


CHAPTER  V 
Intersection  of  Planes  with  Surfaces,  and  the  Development  of  Surfaces 


115.  To  determine  the  intersection  of  any  surface 

with  any  secant  plane         ....       82 

116.  A  tangent  to  a  curve  of  intersection       .         .       83 

1 17.  The  true  size  of  the  cut  section      .        ,        .83 

118.  A  right  section 83 


119.  The  development  of  a  surface         .         .         .83 

120.  To  determine  the  intersection  of  a  plane  with 

a  pyramid .83 

121.  To  develop  the  pyramid  ....       84 

122.  To  determine  the  curve  of  intersection  be- 

tween a  plane  and  any  cone        ...       86 


TABLE  OF  CONTENTS 


IX 


123.   To  determine  the  development  of  any  oblique  128. 

cone '      .         .86      129. 

12-1.    To  determine  the  curve  of  intersection  be-  130. 

tweeu  a  plane  and  any  cylinder ...       88 

125.  To  develop  the  cylinder  .         .         .         .       88      131. 

126.  The  development  of  a  cylinder  when  the  axis  132. 

is  parallel  to  a  coordinate  plane  .         .       91 

127.  To  determine  the  curve  of  intersection  be- 

tween a  plane  and  a  prism  ...       92 


To  develop  the  prism 

The  Helical  Convolute 

To  draw  elements  of  the  surface  of  the  heli- 
cal convolute 

To  develop  the  helical  convolute    . 

To  determine  the  curve  of  iutersection  be- 
tween a  plane  and  a  surface  of  revolu- 
tion      . 


93 
93 

94 
95 


96 


CHAPTER   VI 

IXTERSECTIOX    OF    SURFACES 


133.  General  principles 99 

134.  Character  of  Auxiliary  Cutting  Surfaces       .       99 

135.  To  determine  the  curse  of  intersection  be- 

tween a  cone  and  cylinder  with  axes  oblique 

to  the  coordinate  plane        ....     100 

136.  Order  and  Choice  of  Cutting  Planes      .         .     103 

137.  To  determine  if  there  be  one  or  two  curves 

of  iutersection 103 

138.  To  determine   the   visible  portions   of   the 

curve 104 

139.  To  determine  the  curve  of  intei-section  be- 

tween two  cylinders,  the  axes  of  w  hich  are 
oblique  to  the  coordinate  planes         .        .     104 


140.  To  determine  the  curve  of  intersection  be- 

tween two  cones,  the  axes  of  which  are 
oblique  to  the  coordinate  planes 

141.  To  determine  the  curve  of  intersection  be- 

tween an  ellipsoid  and  an  oblique  cylinder 

142.  To  determine  the  curve  of  intersection  be- 

tween a  torus  and  a  cylinder,  the  axes  of 
which  are  pei-pendicular  to  the  horizontal 
coordinate  plane 

143.  To  determine  the  curve  of  intersection  be- 

tween an  ellipsoid  and  a  paraboloid,  the 
axes  of  which  intersect  and  are  parallel  to 
the  vertical  coordinate  planes    . 


CHAPTER  Vn 
Warped  Surfaces 


144.  "Warped  Surface.     Classification    .         .         .     109 

145.  Having   given   three   curvilinear  directrices 

and  a  point  on  one  of  them,  it  is  required 


to  determine  the  two  projections  of  the 
element  of  the  warped  surface  passing 
through  the  given  point    .... 


104 
105 


106 


108 


110 


TABLE  OF  CONTENTS 


146.  Having  given  two  curvilinear  directrices  and 

a  plane  director,  to  draw  an  element  of 
the  warped  surface,  Case  1  .         .         . 

147.  Having  given  two  curvilinear  directrices  and 

a  plane  director,  to  draw  an  element  of 
the  warped  surface.  Case  2  .         .         . 

148.  Modifications  of  the  types  in  Arts.  145  and 

146 

149.  The  Hyperbolic  Paraboloid    .... 

150.  Through  a  given  point  on  a  directrix  to  draw 

an  element  of  the  hyperbolic  paraboloid    . 

151.  Having  given  one  projection  of  a  point  on  an 

hyperbolic  paraboloid,  to  determine  the 
other  projection,  and  to  pass  an  element 
through  the  point 


»AOE 

ART. 

152. 

153. 

111 

154. 

155. 

156, 

112 

157. 

112 

114 

158. 

116 

159. 

160. 

116 


PAGB 

Warped  Helicoids 118 

Right  helicoid 118 

General  type  of  warjied  helicoids  .  .  .118 
Hyperboloid  of  Revolution  of  one  Nappe  .  118 
Through  any  point  of  the  surface  to  draw 

an  element 120 

The  Generatrix  may  be  governed  by  Three 

Rectilinear  Directrices        ....     121 
The  Generatrix  may  be  governed  by  Two 

Curvilinear  Directrices  and  a  cone  Director     121 
Tangent  plane  to  an  hyperboloid  of  revolu- 
tion    .         .         .       ' 121 

Through  a  right  line  to  pass  a  plane  tangent 
to  any  double-curved  surface  of  revolution. 
General  solution 122 


161.   Directions  for  solving  the  problems 


CHAPTER  Vni 
Problems 
.     124      162.    Problems 


125 


DESCRIPTIVE    GEOMETRY 


OF   THE 

UNIVERSl^  ' 

OF 


DESCRIPTIVE    GEOMETRY 


CHAPTER   I 


DEFINITIONS  AND  FIRST  PRINCIPLES 


I.  Descriptive  Geometry  is  the  art  of  graph- 
ically solviug  problems  involving  three  dimen- 
sions. By  its  use  the  form  of  an  object  may 
be  graphically  defined,  and  the  character, 
relation,  and  dimensions  of  its  lines  and  sur- 
faces determined. 

To  the  student  it  presents  the  most  admir- 
able training  in  the  use  of  the  imagination, 
such  as  the  engineer  or  architect  is  called  upon 
to  exercise  for  the  development  of  new  forms 
in  structure  and  mechanism,  and  which  must 
be  mentally  seen  before  being  graphically  ex- 
pressed. 

To  the  engineer  and  architect  it  supplies 
the  principles  for  the  solution  of  all  problems 
relating  to  the  practical  representation  of  forms. 


as  illustrated  by  the  various  types  of  working 
drawiugs  which  are  used  by  artisans  to  exe- 
cute designs. 

It  is  the  foundation  for  the  understanding 
of  the  different  systems  of  projection  such  as 
orthographic,  oblique,  and  perspective. 

2.  Projection.  The  representation  of  an 
object  is  made  on  one  or  more  planes  by  a 
process  known  as  projection,  the  picture,  draw- 
ing, or  projection  of  the  object  being  deter- 
mined by  the  intersection  of  a  system  of  lines 
with  the  plane.  The  lines  are  known  as  pro- 
jectors and  are  drawn  from  the  ojaect  to  the 
plane  of  projection,  or  picture  plane^  If  these 
lines,  or  projectors,  are  perpendicular  to  the 
plane  of  projection,  the  system  is  known  as 


DESCRIPTIVE  GEOMETRY 


Orthographic  Projection,  arid  at  least  two  pro- 
jections, or  views,  are  required  to  fully  repre- 
sent the  object,  Fig.  1  represents  two  views 
of  a  box  by  orthographic  projection.  This 
system  is  the  one  commonly  employed  for 
working  drawings,  and  for  the  solution  of 
problems  in  Descriptive  Geometry. 

If  the  projectors  are  parallel  to  each  other, 
and  oblique  to  the  plane  of  projection,  the 
system  is  known  as  Oblique  Projection.*  Fig.  2 
represents  the  application  of  this  method  to  the 
illustration  of  the  object  shown  in  Fig.  1.  This 
system  is  used  for  the  purpose  of  producing  a 
more  pictorial  effect,  but  one  which  is  easily 
executed  and  susceptible  of  measurement. 

If  the  projectors  converge  to  a  point  on  the 
opposite  side  of  the  plane  of  projection,  the 
system  is  known  as  Perspective.  Fig.  3  is  an 
application  of  this  method  to  the  representa- 
tion of  the  object  previously  illustrated.  This 
system  is  used  to  produce  the  pictorial  effect 

*For  a  treatise  on  Oblique  Projection,  and  that  branch 
of  Orthographic  Projection  known  as  Isometric  Projection, 
see  " Elements  of  Mechanical  Drawing"  of  this  series. 


obtained  by  the  camera,  and  is  chiefly  em- 
ployed by  architects  for  the  representation  of 
buildings  as  they  will  appear  to  the  eye  of  an 
observer. 

3.  Coordinate  Planes.  The  planes  upon 
wliich  the  representations  are  made  are  called 
coordinate  planes,  or  planes  of  projection,  and 
are  usually  conceived  to  be  perpendicular  to 
each  other  and  indefinite  in  extent.  Fig.  4 
shows  their  relative  positions,  but  for  conven- 
ience of  representation  they  are  limited  in 
extent.  The  plane  designated  by  5"  is  called 
the  horizontal  coordinate  plane,  and  the  repre- 
sentation made  thereon  is  known  as  the  hori- 
zontal projection,  plan,  or  top  vieiv.  The  plane 
designated  by  F'is  called  the  vertical  coordinate 
plane,  and  the  representation  made  thereon  is 
known  as  the  vertical  projection,  elevatioti,  or 
front  view.  The  plane  designated  by  P  is 
called  the  profile  coordinate  plane,  and  the 
representation  made  thereon  is  known  as  the 
profile  projection,  side  elevation,  end  or  side 
view.  The  line  of  intersection  between  the  V 
and  ^planes  is  known  as  the  ground  line. 


ORTHOGRAPHIC  PROJECTION 


4.  Quadrants  or  Angles.  The  poi-tion  of 
space  lying  in  each  of  the  four  diedral  angles 
formed  by  the  vertical  and  horizontal  coordi- 
nate planes  is  designated  as  follows  : 

1st  quadrant,  above  H  and  before  V. 
2nd  quadrant,  above  R  and  behind  V. 
3rd  quadrant,  below  ^and  behind  V. 
4th  quadrant,  below  H  and  before  V. 

5.  Orthographic  Projection.  The  projection* 
of  a  point  on  any  coordinate  plane  is  obtained 
by  letting  fall  a  perpendicular  from  the  point 
in  space  to  the  coordinate  plane,  its  intersec- 
tion with  that  plane  being  the  projection  of 
the  point.  The  perpendicular  is  called  the 
projector  or  projecting  line.  In  Fig.  4,  a  is  the 
point  in  space  and  its  projections  are  designated 
by  the  same  letter  with  r,  A,  or  p  written 
above  and  to  the  right,  as  a%  signifying  the 
vertical  projection,  a^,  the  horizontal  projec- 
tion, and  flP,  the  profile  projection. 

For  convenience  of  representation,  the  verti- 
cal coordinate  plane  is  conceived  as  revolved 

*  "  Projection "  used  without  a  qualifying  adjective 
always  means  orthographic  projection. 


HORIZONTAL  PROJECTION 


VERTICAL  PROJECTION 
Fig.   I. 


OBLIQUE  PROJECTION 
Fig.  2. 


DESCRIPTIVE  GEOMETRY 


about  the  ground  line  to  coincide  with  the  hori- 
zontal coordinate  plane  in  such  a  way  that  the 
first  and  third  quadrants  will  be  opened  to 
180°,  and  the  second  and  fourth  quadrants  will 
be  closed  to  0°.  In  comparing  Figs.  5,  6,  and 
7  it  will  be  observed  that  any  point  in  the 
fourth  quadrant,  as  point  a,  has,  after  the 
planes  have  been  folded  together,  both  pro- 
jections below  the  ground  line;  any  point  in 
the  third  quadrant,  as  point  6,  has  the  vertical 
projection  below,  and  the  horizontal  projection 
above,  the  ground  line  ;  any  point  in  the  second 
quadrant,  as  point  (?,  has  both  projections  above 
the  ground  line;  and  any  point  in  the  first 
quadrant,  as  point  d,  has  the  vertical  projection 
above,  and  the  horizontal  projection  below,  the 
ground  line.  Thus  in  P'ig.  7  it  is  evident  that 
the  portion  of  the  paper  above  the  ground  line 
represents  not  only  that  part  of  the  horizontal 
codrdinate  plane  which  lies  behind  the  vertical 
coordinate  plane,  but  also  that  part  of  the  ver- 
tical coordinate  plane  which  lies  above  the 
horizontal  coordinate  plane.  Likewise  the 
paper  below  the  ground  line  represents  that 


portion  of  the  horizontal  coordinate  plane  ly- 
ing in  front  of  the  vertical  coordinate  plane, 
and  also  that  portion  of  the  vertical  coordinate 
plane  lyiug  below  the  horizontal  coordinate 
plane. 

The  profile  coordinate  plane  is  commonly 
revolved  about  Gr^L  nsan  axis  until  it  coincides 
with  the  vertical  coordinate  plane,  as  in  Figs. 
8  and  9;  but  it  may  be  revolved  about  Gr^  L  as 
an  axis  until  it  coincides  with  the  horizontal 
coordinate  plane,  as  in  Figs.  10  and  11.  It 
makes  no  difference  as  to  the  correct  solution 
of  the  problem  whether  the  profile  coordinate 
plane  be  revolved  to  the  right  or  to  the  left, 
but  care  should  be  used  to  revolve  it  into  such 
a  position  as  will  cause  the  least  confusion  with 
other  lines  of  the  problem. 

6.  Notation.  Points  in  space  are  designated 
by  small  letters,  as  a,  b,  c,  etc.,  and  their  vertical, 
horizontal,  and  profile  projections  by  the  same 
letters  with  v,  A,  and  p  placed  above  and  to  the 
right,  as  a*,  a'',  a^.  When  revolved  into  one 
of  the  coordinate  planes,  the  points  will  be 
designated  by  a',  i',  or  a",  6",  etc. 


CXX>RDINATE  PLANES 


ORTHOGHAPHIC  PROJECTION 

Fig.  7. 


Fig.  I  I , 


6 


DESCRIPTIVE   GEOMETRY 


A  line  in  space  is  designated  by  two  of  its 
points,  as  ah,  or  by  a  capital  letter,  usually 
one  of  the  first  ten  of  the  alphabet,  as  A,  and 
its  projections  are  designated  by  a^'J",  a'^U', 
aPlP,  or  A\  A\  A^.  When  revolved  into  one 
of  the  coordinate  planes,  the  lines  will  be  desig- 
nated by  a'h\  a"h",  or  A',  A",  etc.  The  trace 
of  a  line,  i.e.  the  point  in  which  it  pierces  a 
cocirdinate  plane,  may  be  designated  by  V-tr, 
Htr,  or  P-tr,  according  as  the  line  pierces  the 
vertical,  horizontal,  or  profile  coordinate  plane. 

A  plane  in  space  is  determined  by  three 
points  not  in  the  same  straight  line ;  by  a  line 
and  a  point ;  or  by  two  parallel  or  intersecting 
lines.  In  projection  a  plane  is  usually  desig- 
nated by  its  traces,  i.e.  the  lines  in  which  it 
pierces  the  coordinate  planes.  These  traces 
are  designated  by  the  last  letters  of  the  alpha- 
bet beginning  with  M,  as  VM,  HM,  PM,  accord- 
ing as  the  plane  M  intersects  the  vertical, 
horizontal,  or  profile  coordinate  plane. 

Abbreviations.  The  following  abbreviations 
are  used  in  connection  with  the  figures  and 
problems : 


V,  the  vertical  coordinate  plane. 

If,  the  horizontal  coordinate  plane. 

P,  the  profile  coordinate  plane. 

CrL,  ground   line,    the  line  of  intersection 
between  F'and  IT. 

VP,  the  intersection  between  T'^and  P. 

HP,  the  intersection  between  ^and  P. 

IQ,  2Q,  3Q,  4Q,  the  first  quadrant,  second 
quadrant,  etc. 

In  general,  an  object  in  space  is  definitely 
located  by  two  projections  only,  usually  the 
vertical  and  the  horizontal.  Hereafter,  only 
these  two  projections  Avill  be  considered,  unless 
otherwise  indicated. 

The  character  of  the  lines  employed  is  as 
follows : 

Given  and  required  lines. 

Invisible  and  projection  lines. 

. All  lines  not  designated  above. 

7.  Points.  Two  projections  are  necessary 
to  locate  a  point  with  reference  to  V  and  H. 
From  Figs.  12  and  13  it  is  obvious  that  these 
two  projections  must  always  lie  on  a  perpen- 
dicular to  the  ground  line  after   V  has  been 


POINTS 


revolved  to  coincide  with  H.  The  distance 
from  the  point  in  space  to  H  is  equal  to  the 
distance  from  its  vertical  projection  to  the 
ground  line,  and  the  distance  from  the  point 
in  space  to  T"is  equal  to  the  distance  from  its 
horizontal  projection,  to  the  ground  line.  A 
point  on  either  coordinate  plane  is  its  own 
projection  on  that  plane,  and  its  other  projec- 
tion is  in  the  ground  line,  as  points  c,  c?,  and  e. 
Figs.  12  and  13. 

The  points  represented  in  Figs.  12  and  13 
are  described  as  follows: 

Point  rt,  in  5^,  5  units  from   T''and  8  units 

from  H. 
Point  5,  in  5^,  4  units  from   V  and  7  units 

from  H. 
Point  /,  in  4Q,  2  units  from  V  and.  6  units 

from  H. 
Point  c,  in  JT,  between  IQ  and  4Q^  A:  units 

from  X'. 
Point  rf,  in  T'i  between  IQ  and  2Q,  1  units 

from  ff. 
Point  e,  in  GL. 


8.  Lines.  Since  a  line  in  space  is  deter- 
mined by  any  two  of  its  points,  the  projections 
of  these  two  points  determine  the  projections 
of  the  line.  Figs.  14  and  15.  A  line  may  also 
be   projected    by   passing    planes   through  it 


Fig.  12. 


'a* 
I 


A 


f 

Fig.  13. 

aCAL£ 

lllllllilllllllll 


8 


DESCRIPTIVE  GEOMETRY 


perpendicular  to  the  coordinate  planes.  The 
intersections  of  these  planes  with  V,  JI,  and  P 
will  determine  respectively  the  vertical,  hori- 
zontal, and  profile  projections  of  the  line. 
Such  auxiliary  planes  are  known  us  plane  pro- 
jectors or  projecting  planes  of  the  lines.  •  In 
-Pig.  14  the  plane  ahh^a^  is  the  vertical  project- 
ing plane  of  the  line  ah;  the  plane  abb''a^  is 
the  horizontal  projecting  plane  of  the  line  a5, 
and  the  plane  abb^a^  is  the  profile  projecting 
plane  of  the  line  ab. 

g.  A  line  parallel  to  a  coordinate  plane 
will  have  its  projection  on  that  plane  parallel 
to  the  line  in  space,  and  its  other  projection 
will  be  parallel  to  the  ground  line.  Li^he  A, 
Figs.  16  and  17,  is  a  line  parallel  to  V.  A"  is 
parallel  to  A  in  space,  and  A''  is  parallel  to  GrL. 
Line  B  is  parallel  to  ff,  line  C  is  parallel  to 
both  F'and  IT,  and  line  D  is  parallel  to  P. 

10.  A  line  perpendicular  to  a  coordinate 
plane  will  have  for  its  projection  on  that  plane, 
a  point,  and  its  other  projection  will  be  per- 
pendicular to  the  ground  line.  Line  JE,  Figs. 
18  and  19,  is  a  line  perpendicular  to  V.     U"  is 


a  point  and  U''  is  perpendicular  to  GrL.  Line 
Fis  perpendicular  to  ^,  and  line  ^is  perpen- 
dicular to  P,  K"  and  K''  being  perpendicular 
to  VP  and  IfP,  respectively,  and  JC^  being  a 
point. 

11.  Aline  lying  in  either  coordinate  plane 
is  its  own  projection  on  that  plane,  and  its 
)ther  projection  is  in  the  ground  line.  Jn 
Figs.  20  and  21,  line  A  lies  in  ff  and  line  B 
lies  in  V. 

12.  A  line  parallel  to  one  coordinate  plane 
and  oblique  to  the  other  has  its  projection  on 
the  plane  to  which  it  is  parallel  equal  to  the 
true  length  of  the  line  in  space,  and  the  angle 
which  this  projection  makes  with  the  ground 
line  is  the  true  size  of  the  angle  which  the  line 
makes  with  the  plane  to  which  it  is  oblique. 
Line  A,  Figs.  16  and  17,  is  seen  in  its  true 
length  in  its  vertical  projection,  and  it  makes 
an  angle,  of  30°  with  II. 

13.  If  two  lines  are  parallel  in  space,  their 
projections  will  be  parallel,  Figs.  22  and  23. 
Lines  (7  and  D  are  ]iarallel;  therefore  C  and 
D'  are  parallel,  and  C^  and  D''  are  parallel. 


LINES 


Fig.  17 


Fig.  18. 


J^\ 

A' 

1              > 

:  «* ! 

Fig.  20. 


Fig.  22. 


Fig.  19 


A' 


'       «*    I 


Fig.  21. 


Fig.  23. 


10 


DESCRIPTIVE   GEOMETRY 


t^.  If  two  lines  intersect  in  space,  they 
have  one  point  in  common;  hence,  the  projec- 
tions of  the  lines  will  intersect  each  other  in 
the  projections  of  the  point,  as  at  a*  and  a\ 
Figs.  24  and  25.  If  the  projections  of  the 
lines  do  not  intersect  on  a  common  perpendicu- 
lar to  the  ground  line,  the  lines  in  space  do 
not  intersect.  Lines  A  and  B,  Fig.  26,  do 
not  intersect. 

15.  If  a  line  intersects  the  ground  line,  as 
in  Fig.  27,  its  projections  will  intersect  the 
ground  line  in  the  same  point. 

16.  The  traces  of  a  line  are  the  points  in 
which  the  line  pierces  the  coordinate  planes. 
In  Figs.  28  and  29,  a  is  the  horizontal  trace, 
and  b  the  vertical  trace,  of  line  C.  The  ver- 
tical projection  of  the  horizontal  trace  is  in 
the  ground  line,  as  is  likewise  the  horizontal 
projection  of  the  vertical  trace. 

17.  For  convenience  of  expression,  it  is  cus- 
tomary to  define  the  position  of  a  line  with 
respect  to  the  coordinate  planes  by  giving  the 
quadrant  in  which  it  lies,  together  with  its 


inclination  with,  and  distance  from,  the  coordi- 
nate planes.  Line  (7,  Figs.  28  and  29,  lies  in 
the  first  quadrant,  and  if  read  from  a  toward 
b,  inclines  upward,  backward,  and  toward  the 
right.  The  vertical  projection  indicates  that 
the  inclination  is  upward;  the  horizontal  pro- 
jection indicates  that  the  inclination  is,  at  the 
same  time,  backward;  while  either  projection 
indicates  that  the  inclination  is  to  the  right. 
The  angle  of  inclination  will  be  considered 
later.  If  the  line  be  read  from  b  toward  a,  the 
inclination  would  be  downward,  forward,  and 
to  the  left.  Either  is  correct.  Fig.  30  illus- 
trates four  other  lines  as  follows  : 

Line  ab,  in  3Q,  inclined  upward,  forward, 
and  to  tlie  right. 

Line  cd,  in  2Q,  inclined  downward,  back- 
ward, and  to  the  right. 

Line  ef,  in  4Q,  parallel  to  IT,  inclined  for- 
ward and  to  tlie  right. 

Line  gk,  in  3Q,  parallel  to  P,  and  inclined 
upward  and  forward. 


LINES 


11 


Fig.  26. 


12 


DESCRIPTIVE  GEOMETRY 


i8.  Planes.  The  position  of  planes  may  be 
represented  in  projection  as  follows: 

1.  By  the  projections  of  two  intersecting  or 
parallel  lines. 

2.  By  the  projections  of  a  line  and  a  point. 

3.  By  the  projections  of  three  points  not  in 
the  same  straight  line. 

4.  By   the   lines   of   intersection  with  the 
coordinate  planes.     (Traces.) 

All  planes  being  indefinite  in  extent  must 
intersect  one  or  botli  of  the  coordinate  planes. 
Such  lines  of  intersection  are  called  the  traces 
of  the  planes.  Fig.  31  illustrates  the  inter- 
sections of  the  planes  N  with  V  and  ff,  the 
vertical  and  horizontal  traces  being  lettered 
FTVand  JIN^,  respectively.  The  orthographic 
representation  is  shown  in  Fig.  32,  save  that 
it  is  not  always  customary  to  continue  the 
traces  beyond  the  ground  line,  the  horizontal 
trace  being  drawn  on  one  side  and  the  vertical 
trace  on  the  other,  as  in  Fig.  40. 

Since  the  vertical  trace  of  a  plane  is  a  line 
lying  on  V,  it  may  also  be  lettered  as  the  verti- 
cal projection  of   a  line.     Thus,  in   Figs.  31 


and  32,  VN  may  be  lettered  A^  and,  according 
to  Art.  11,  Page  8,  A'*  must  coincide  with  CrL. 
Likewise,  JIN  is  a  line  lying  in  II  and  may 
be  lettered  B'',  while  B^  must  lie  in  CrL. 

The  following  positions  of  planes  are  illus- 
trated by  Figs.  33  to  48  inclusive: 

Perpendicular  to  JSTand  parallel  to  V,  Figs. 

33  and  34. 
Perpendicular  to  P^and  parallel  to  ff,  Figs. 

35  and  36. 
Inclined  to    F'and  H,  but  parallel  to   GrL^ 

Figs.  37  and  38. 
Inclined  to  ff  and  perpendicular  to  FJ  Figs. 

39  and  40. 
Inclined  to  F'and  perpendicular  to  ff,  Figs. 

41  and  42. 
Perpendicular  to  V  and  H,  Figs.  43  and  44. 
Inclined  to  F'and  IT,  but  containing    CrL, 

Figs.  45  and  46. 
Inclined  to  V,  H,  and  GrL,  Figs.  47  and  48. 
The  traces  of  parallel  planes  are  parallel. 
19.   From  the  foregoing  illustrations  it  will 
be  observed  that  the  ground  line  is  the  hori- 
zontal  projection    of   the   vertical   coordinate 


PLANES 


13 


plane,  and  that  any  point,  line,  or  plane  lying 
in  V  will  have  its  entire  horizontal  projection 
in  the  ground  line.  Likewise,  it  will  be  ob- 
served that  the  ground  line  is  the  vertical 
projection  of  the  horizontal  coordinate  plane, 
and  that  any  point,  line,  or  plane  lying  in  ff 
will  have  its  entire  vertical  projection  in  the 
ground  line. 


Fig.  33. 


Fig.  37. 


HM 


Fig.  34. 


HM 


VN 


Fig.  38. 


Fig.  41.  Fig.  42. 


YN 


Fig.  35.  Fig.  36. 


% 

X 

^\ 

^ 

/ 

y 

Fig.  39.  Fig.  40. 


Fig.  43.  Fig.  44. 


Fig.  32. 


Fig.  47.  Fig.  48. 


CHAPTER   II 


POINTS,  LINES,  AND  PLANES 


20.  Three  distinct  operations  are  required 
for  the  solution  of  problems  in  Descriptive 
Geometry. 

First,  a  statement  of  the  Principles  in- 
volved. 

Second,  an  outline  of  the  Method  to  be 
observed,  by  the  enumeration  of  the  steps 
necessitated. 

Third,  the  graphic  Construction  of  the 
problem.  The  first  two  operations  are  purely 
mental,  and  the  last  is  the  mechanical  opera- 
tion of  executing  the  drawing. 

21.  To  determine  three  projections  of  a  line. 
Principle.     The  projections  of  two  points 

of  a  line  determine  the  projections  of  the  line. 
Method.  1.  Determine  the  vertical,  hori- 
zontal, and  profile  projections  of  two  points  of 
the  line.  2.  Connect  the  vertical  projections 
of  the  points  to  obtain  the  vertical  projection  of 


the  line ;  connect  the  horizontal  projections 
of  the  points  to  obtain  the  horizontal  projec- 
tion of  the  line ;  and  connect  the  profile  pro- 
jections of  the  points  to  obtain  the  profile 
projection  of  the  line. 

Construction.  Figs.  49  and  50.  Let  it 
be  required  to  determine  the  projections  of  a 
line  passing  through  point  a,  in  IQ,  4:  units 
from  F",  7  units  from  If,  and  through  point  b, 
in  4Q,  16  units  from  V,  9  units  from  IT;  point 
a  to  be  12  units  to  the  left  of  point  b.  On 
any  perpendicular  to  GL  lay  off  J",  9  units 
below  GrL  and  b\  16  units  below  GL  (Art.  7, 
page  6).  On  a  second  perpendicular,  12  units 
to  the  left  of  b"  and  b'^,  lay  off  a",  7  units  above 
GL  and  a\  4  units  below  GL.  Connect  a" 
and  b"  to  obtain  the  vertical  projection  of  the 
line,  and  connect  a''  and  b'^  to  obtain  the  hori- 
zontal projection  of  the  line. 


U 


PROJECTIONS  OF  A  LINE 


15 


To  obtain  a^  and  J^,  assume  the  position  of 
the  profile  plane,  shown  in  Fig.  49  by  its  inter- 
section with  V  and  ff  as  VP  and  ffP.  Re- 
volve P  about  VP  as  an  axis  to  coincide  with 
V.  Then  a^  will  lie  on  a  line  drawn  through 
a"  parallel  to  CrL,  and  at  a  distance  from  VP 
equal  to  that  of  a''  from  GL.  Tliis  is  obtained 
bj  projecting  a''  to  HP  and  revolving  HP 
about  X  as  a  center  to  coincide  with  GL,  and 
projecting  perpendicularl}^  to  meet  the  parallel 
to  GL  through  a"  at  a^.  Obtain  b^  in  like 
manner.  Connect  a''  and  b^  to  obtain  the  pro- 
file projection  of  the  line. 

22.  If  point  e  in  2Q  be  one  of  the  given 
points,  ^  and  e"  having  been  determined, 
obtain  e^  as  follows:     Project  e''  to  HP,  re- 


volve ffP  to  GL,  and  project  perpendicularly 
to  meet  a  line  drawn  parallel  to  GL  through 
e"  at  «**;  but  after  e*  has  been  projected  to  HP 
it  is  imperative  that  HP  be  revolved  in  the 
same  direction  that  it  was  revolved  when  the 
profile  projections  of  the  other  points  were  de- 
termined. If  the  horizontal  projection  of  one 
point  be  revolved,  then  the  horizontal  projec- 
tions of  all  points  must  be  revolved,  and  ver- 
tical projections  must  not  be  revolved. 

23.  P  may  be  revolved  in  either  of  the 
directions  shown  by  Figs.  49  and  51.  Fig.  51 
represents  the  line  when  P  has  been  revolved 
about  HP  as  an  axis  to  coincide  with  H. 
Here  it  vdll  be  observed  that  the  vertical  pro- 
jections, and  only  these,  have  been  revolved. 


Fig.  49 


Fig.  51. 


16 


DESCRIPTIVE  GEOMETRY 


24.     To  determine  the  traces  of  a  line. 

Principle.  The  traces  of  a  line  are  the 
points  in  which  the  line  pierces  the  coordinate 
planes.  The  projections  of  these  traces  must, 
therefore,  lie  in  the  projections  of  the  line,  and 
one  projection  of  each  trace  will  lie  in  the 
ground  line  (Art.  7,  page  6). 

Method.  1.  To  obtain  the  vertical  trace 
of  the  line,  continue  the  horizontal  projection 
of  the  line  until  it  intersects  the  ground  line ; 
this  will  be  the  horizontal  projection  of  the 
vertical  trace,  and  its  vertical  projection  will 
be  perpendicularly  above  or  below  the  ground 
line  in  the  vertical  projection  of  the  given 
line.  2.  To  obtain  the  horizontal  trace  of 
the  line,  continue  the  vertical  projection  of  the 
line  until  it  intersects  the  ground  line ;  this 
will  be  the  vertical  projection  of  the  horizon- 
tal trace,  and  its  horizontal  projection  will 
be  vertically  above  or  below  the  ground 
line  in  the  horizontal  projection  of  the  given 
line. 

Case  1.  When  the  line  is  inclined  to  V, 
H,  and  P. 


Construction.  Figs.  52  and  53.  Let  it 
be  required  to  determine  the  vertical  trace  of 
line  A.  Continue  A^  until  it  intersects  GL 
in  c*,  which  is  the  horizontal  projection  of  the 
vertical  trace.  Next  project  this  point  to  yl", 
as  at  c",  wliich  is  the  vertical  projection  of  the 
vertical  trace. 

The  horizontal  trace  is  similarly  determined 
thus  :  Continue  A^  until  it  intersects  GL  in 
d",  which  is  the  vertical  projection  of  the  hori- 
zontal trace.  Next  project  this  point  to  A^,, 
as  at  c?'',  which  is  the  horizontal  projection  of 
the  horizontal  trace. 

To  determine  the  profile  trace,  consider  P, 
Fig.  52,  to  be  represented  by  its  intersections 
with  F'and  H^  as  VP  and  HP^  and  to  be  re- 
volved to  the  right  about  VP  as  an  axis  until 
it  coincides  with  V.  Continue  A^  until  it 
intersects  VP  in/",  which  is  the  vertical  pro- 
jection of  the  profile  trace.  Continue  A^  until 
it  intersects  HP  in/'',  which  is  the  horizontal 
projection  of  the  profile  trace,  revolve  to  6ri, 
using  the  intersection  of  HP  with  CrL  as  a 
center,  and  project   to  f^  by   a   line    drawn 


TRACES  OF  A   LINE 


17 


through /"  parallel  to  CrL.     This  is  the  profile 
projection  of  the  profile  trace. 

Fig.  53  is  the  oblique  projectioa  of  the  line, 
and  clearly  shows  that  the  line  passes  from 
one  quadrant  to  another  at  its  vertical  and 
horizontal  traces,  i.e.  line  A  passes  from  IQ 
to  2Q  at  its  vertical  trace,  c,  and  from  i^  to 


4Q  at  its  horizontal  trace,  d. 

XoTE.  The  vertical  projection  of  the  verti- 
cal trace  of  a  line  is  often  called  the  vertical 
trace  of  the  line,  since  this  trace  and  its  ver- 
tical projection  are  coincident.  Likewise  the 
horizontal  projection  of  the  horizontal  trace  is 
called  the  horizontal  trace  of  the  line. 


Fig.  521 


18 


DESCRIPTIVE   GEOMETRY 


25.  Case  2.  When  the  line  is  inclined  to 
V  and  H  and  is  parallel  to  P. 

Construction.  Fig.  54.  If  the  given  line 
is  parallel  to  P,  its  vertical  and  horizontal 
traces  cannot  be  determined  by  the  above 
method,  since  the  projections  of  the  line  are 
perpendicular  to  the  ground  line;  hence,  a 
profile   projection   of   the   line    is   necessary. 


T*?"  Fig.  54. 

Let  it  be  required  to  determine  the  vertical 
and  horizontal  traces  of  the  line  ab.  P  is 
assumed  at  will  and  is  indicated  by  HP  and 
VP.  Determine  a^b^,  the  profile  projection  of 
the  line  (Art.  21,  page  14),  by  revolving  P 
about  VP  as  an  axis  until  it  coincides  with  V. 


That  portion  of  P  which  before  revolution 
was  in  5^  now  falls  below  GrL  and  to  the 
right  of  VP ;  that  portion  which  was  in  4Q 
falls  below  GrL  and  to  the  left  of  VP;  that 
portion  which  was  in  i  ^  falls  above  CrL  and 
to  the  left  of  HP.  Tlius  the  profile  projec- 
tion of  the  line,  when  continued,  indicates 
that  the  line  passes  through  3Q  into  2Q  lit 
point  df,  d^  being  the  profile  projection  of  the 
horizontal  trace.  Likewise  the  line  passes 
from  5^  into  4Q  at  point  c,  c^  being  the  profile 
projection  of  the  vertical  trace.  Counter- 
revolve  P  to  its  original  position  and  obtain 
dl"  and  d",  which  are  the  horizontal  and  vertical 
projections  of  the  horizontal  trace,  and  c"  and 
c'',  which  are  the  vertical  and  horizontal  pro- 
jections of  the  vertical  trace.  Since  line  ab  is 
parallel  to  P,  it  has  no  profile  trace. 

26.  To  determine  the  projections  of  a  line 
when  its  traces  are  given  (Art.  21,  page  14). 

Method.  1.  Determine  the  horizontal 
projection  of  tha  vertical  trace,  and  the  ver- 
tical projection  of  the  horizontal  trace. 
2.  Connect  the  horizontal  trace  with  the  hori- 


LINES  LYING   IN  PLANES 


19 


zontal  projection  of  the  vertical  trace  to  obtain 
the  horizontal  projection  of  the  line.  Connect 
the  vertical  trace  with  the  vertical  projection 
of  the  horizontal  trace  to  obtain  the  vertical 
projection  of  the  line. 

27.  Conditions  governing  lines  lying  in  a 
plane.  Since  tiie  traces  of  a  plane  are  lines  of 
the  plane,  they  must  intersect  all  other  lines  of 
the  plane,  and  conversely,  all  lines  of  a  plane 
must  have  their  traces  in  the  traces  of  the 
plane.     See  lines  \A  and  B^  Figs.  55  and  56. 


28.  If  a  line  is  parallel  to  H^  jt-^ntersects 
H  at  infinity;  therefore,  its  horizontal  trace  is 
at  infinity  and  its  horizontal  projection  is 
parallel  to  the  horizontal  trace  of  the  plane  in 
which  it  lies,  its  vertical  projection  being 
parallel  to  the  ground  line.  See  line  C,  Figs. 
57  and  58.  Likewise  if  a  line  is  parallel  to  V^ 
its  vertical  projection  is  parallel  to  the  vertical 
trace  of  the  plajie  in  which  it  lies,  and  its 
horizontal  projection  is  parallel  to  the  ground 
line. 


Fig.  56 


Fig   57. 


Fig.  58. 


20 


DESCRIPTIVE  GEOMETRY 


29.  If  the  traces  of  any  plane  be  drawn 
through  the  traces  of  a  line,  the  plane  must 
contain  the  line  ;  therefore,  an  infinite  number 
of  planes  may  be  passed  through  any  line.  In 
Fig.  59,  planes  iV,  R,  T,  and  S  all  contain  line 
E. 

30'.  To  pass  a  plane  through  two  intersect- 
ing or  parallel  lines. 

Pkinciple.  The  traces  of  the  plane  must 
contain  the  traces  of  the  lines  (Art.  27, 
page  19). 

Case  1.  Method.  1.  Determine  the  traces 
of  the  given  lines.  2.  Connect  the  two  liori- 
zontal  traces  of  the  lines  to  obtain  the  horizon- 
tal trace  of  the  plane,  and  connect  the  two 
vertical  traces  of  the  lines '  to  obtain  the 
vertical  trace  of  the  plane. 

Construction.  Fig.  60.  A  and  B  are  the 
given  intersecting  lines.  Determine  their 
horizontal  traces  <^  and  dl^^  and  their  vertical 
traces  e"  and  /"  (Art.  24,  page  16).  Con- 
nect the  horizontal  traces  to  determine  HT 
and  the  vertical  traces  to  determine  VT. 
Since  the  traces  of  the  plane  must  meet  in  GiL., 


only  three  traces  of  the  lines  are  necessary. 

Check.  Both  traces  of  the  plane  must 
intersect  the  ground  line  in  the  same  point. 

Note.  The  vertical  projection  of  the  ver- 
tical trace  of  a  plane  will  always  be  spoken  of 
as  the  vertical  trace  of  the  plane,  but  it  must 
constantly  be  borne  in  mind  that  the  vertical 
trace  of  a  plane  is  a  line  lying  on  V  and  that 
its  horizontal  projection  is  in  the  ground  line. 
Likewise  the  horizontal  projection  of  the  hori- 
zontal trace  of  a  plane  will  be  spoken  of  as  the 
horizontal  trace  of  that  plane,  but,  as  before, 
the  horizontal  trace  is  a  line  lying  in  H  and  its. 
vertical  projection  is  in  the  ground  line. 

31.  Case  2.  Method.  If  the  traces  of 
the  given  lines  cannot  readily  be  found,  new 
lines  intersecting  the  given  lines  may  be 
assumed,  which,  passing  through  two  points 
of  the  plane,  lie  in  it,  and  their  traces,  there- 
fore, are  points  in  the  traces  of  the  required 
plane. 

Construction.  Fig.  61.  The  two  given 
intersecting  lines  are  A  and  B,  the  traces  of 
which  cannot  be  found  within  the  limits   of 


THE  PLANE  OF:  LINES 


21 


the  drawing.  Line  C  is  an  assumed  line  join- 
ing point  e  of  line  A  and  point/  of  line  jB,  the 
traces  of  which  are  easily  located  at  g  and  k. 
Line  P  is  a  second  similar  line.  ST^  connect- 
ing the  horizontal  traces  of  lines  G  and  D,  is 
the  required  horizontal  trace  of  the  plane  of 
lines  A  and  B.  Likewise  YT^  connecting  the 
vertical  traces  of  lines  C  and  i),  is  the  required 
vertical  trace  of  the  plane  of  lines  A  and  B. 
32.     Case  3.    Method.    If  one  of  the  given 


lines  is  parallel  to  the  ground  line,  the  required 
plane  will  be  parallel  to  the  ground  line,  and, 
therefore  (Figs.  37  and  38,  page  13),  the 
traces  of  the  plane  will  be  parallel  to  the 
ground  line.  This  problem  may  be  solved  by 
Case  2,  or  by  the  following  method:  Determine 
the  profile  trace  of  the  required  plane  by  ob- 
taining the  profile  traces  of  the  given  lines. 
Having  found  the  profile  trace  of  the  plane,  de- 
termine the  horizontal  and  vertical  traces. 


Fig.  59 


22 


DESCRIPTIVE  GEOMETRY 


Construction.  Fig.  62.  Let  J.  and  J5  be 
the  given  lines  parallel  to  the  ground  line. 
Assume  P  and  draw  H^  and  VP.  Continue 
the  horizontal  and  vertical  projections  of  lines 
A  and  B  to  intersect  HP  and  FP,  respectively. 
The  horizontal  and  vertical  projections  of  the 
profile  trace  of  line  A  will  lie  at  c*  and  e",  and 
the  profile  trace  at  d\  Similarly  determine  d^^ 
the  profile  trace  of  line  B.  Through  cf  and 
dP  draw  PiV,  the  profile  trace  of  the  required 
plane.  E^  will  be  tlie  profile  projection  of  the 
horizontal  trace  of  tlie  required  plane,  and  Kp 
the  profile  projection  of  the  vertical  trace.  By 
counter-revolution  obtain  ^iVand  VN. 

33.     To  pass  a  plane  through  a  line  and  a 
'  point. 

Method.     Connect   the   given   point  with 
any  assumed  point  of  the  line  and  proceed  as 
in  Art.  30,  page  20. 
"^  34.     To  pass  a  plane  through  three  points 
not  in  the  same  straight  line. 

Method.  Connect  the  three  points  by  aux- 
iliary lines  and  proceed  as  in  Art.  30,  page  20. 
In  Fig.  63,  a,  J,  and  c  are  the  given  points. 


35.  Given  one  projection  of  a  line  lying  on 
a  plane,  to  determine  the  other  projection. 

Principle.  The  traces  of  the  line  must 
lie  in  the  traces  of  the  plane  (Art.  27,  page 
19). 

Method.  Determine  the  traces  of  the  line 
and  from  them  determine  the  unknown  pro- 
jection of  the  line. 

Construction.  Fig.  64.  LetiTTVand  FiV 
be  the  traces  of  the  given  plane,  and  A''  one 
projection  of  line  A  lying  in  iV^.  Continue  A'^ 
to  nieet  HN  in  a\  the  horizontal  trace  of  line 
A;  a"  is  in  GL  (Art.  24,  page  16).  Con- 
tinue A!'  to  meet  GiL  in  5\  the  horizontal  pro- 
jection of  the  vertical  trace;  5",  the  vertical 
trace,  is  in  VN.  Connect  a"  and  5''  to  obtain 
A^\  the  required  vertical  projection  of  line  A. 

If  ^"  had  been  the  given  projection.  A'' 
would  have  been  similarly  determined. 

If  the  given  projection,  J?",  Fig.  65,  be  par- 
allel to  GL,  B"  will  be  parallel  to  HS,  for,  if 
a  line  lying  in  an  inclined  plane  has  one  'pro- 
jection parallel  to  the  ground  line.,  the  other  pro- 
jection is  paralleVto  the  trace  of  the  plane;  and 


PROJECTION  OF  LINES  IN  PLANES 


23 


conversely,  if  a  line  lying  in  an  inclined  plane 
has  one  projection  parallel  to  the  trace  of  the 
plane^  the  other  projection  is  parallel  to  the 
ground  line  (Art.  28,  page  19).  Therefore, 
to  determine  B*"  continue  B"  to  meet  VS  in 
c%  the  vertical  trace  of  the  line.  Its  horizon- 
tal projection,  c^,  will  be  in  GL,  and  B^  will 
pass  through  c''  parallel  to  ^*S'. 


E"                  HN 

X* 

* 

c"     Xc" 

B" 

1/*   1  Xrf" 

1         iXx" 

A" 

c" 

B' 

d' 

^^-^               VN 

Fig.  64 


Fig.  62. 


24 


DESCRIPTIVE  GEOMETRY 


36.  Given  one  projection  of  a  point  lying  on 
a  plane,  to  determine  the  other  projection. 

Principle.  The  recjuired  projection  of 
the  point  will  lie  on  the  projection  of  any  line 
of  the  plane  passed  through  the  point. 

Method.  1.  Through  the  given  projec- 
tion of  the  point  draw  the  projection  of  any 
line  lying  o;i  the  plane.  2.  Determine  the 
other  projection  of  the  line  (Art.  35,  page 
22).  3.  The  required  projection  of  the  point 
will  lie  on  this  projection  of  the  line. 

Construction.  Figs.  66  and  67.  Let  a^ 
be  the  given  projection  of  point  a  on  plane  N. 
Through  a!^  draw  B^^  the  horizontal  projection 
of  any  line  of  plane  N  passing  through  point 
a.  Determine  B"  (Art.  35,  page  22).  Then 
a"  lies  at  the  intersection  of  B^  and  a  per- 
pendicular to  GL  through  a^.  The  same 
result  is  obtained  by  using  line  G  lying  in 
plane  N  and  parallel  to  F^  or  using  line  J) 
lying  in  plane  iVand  parallel  to  H. 

If  the  vertical  projection  of  the  point  be 
given,  the  horizontal  projection  is  determined 
in  a  similar  manner. 


In  general,  solve  the  problem  hy  the  use  of 
an  auxiliary  line  parallel  to  V  or  H. 

37.  To  locate  a  point  on  a  given  plane  at  a 
given  distance  from  the  coordinate  planes. 

Principle.  The  required  point  lies  at  the 
intersection  of  two  lines  of  the  given  plane; 
one  line  is  parallel  to,  and  at  the  given  dis- 
tance from,  F",  and  the  other  line  is  parallel  to, 
and  at  the  given  distance  from,  H. 

Method.  1.  Draw  a  line  of  the  plane 
parallel  to,  and  at  the  required  distance  from, 
one  of  the  coordinate  planes  (Fig.  65,  page 
23).  2.  Determine  a  point  on  this  line  at 
the  required  distance  from  the  other  coordi- 
nate plane. 

Construction.  Fig.  68.  Let  it  be  re- 
quired to  locate  point  h  on  plane  R^  at  x 
distance  from  H  and  y  distance  from  V. 
Draw  line  A  in  plane  R^  parallel  to, 
and  at  y  distance  from,  V.  A^  will  be 
parallel  to,  and  at  y  distance  from,  (ri,  and 
A"  will  be  parallel  to  VR  (Art.  35,  page  22). 
On  A^  determine  J"  at  x  distance  from  6ri, 
and  project  to  A!"  to  determine  h^.     Or,  point 


TO  REVOLVE  A  POINT 


25 


Fig.  68 


h  may  be  located  by  passing  line  C  in  plane  R^ 
parallel  to,  and  at  x  distance  from,  H. 

If  the  plane  be  parallel  to  GL,  use  the  fol- 
lowing method:  1.  Draw  on  the  given  plane 
any  line  oblique  to  V  and  H.  2.  Locate  the 
required  point  thereon  at  the  required  distance 
from  F'and  H. 

38.  To  revolve  a  point  into  either  coordi- 
nate plane. 

Principle.  The  axis  about  which  the  point 
revolves  must  lie  in  the  plane  into  which  the 
point  is  to  be  revolved.  The  revolving  point 
will  describe  a  circle  wliose  plane  is  perpen- 
dicular to  the  axis,  and  whose  center  is  in  the 
axis.  The  intersection  of  this  circle  and  the 
coordinate  plane  is  the  required  revolved  posi- 
tion of  the  point. 

Method.  1.  Through  that  projection  of 
the  given  point  on  the  plane  in  which  the  axis 
lies  draw  a  line  perpendicular  to  the  axis. 
2.  On  this  perpendicular  lay  off  a  point  hav- 
ing its  distance  from  the  axis  equal  to  the 
hypotenuse  of  a  rigiit  triangle,  one  leg  of  which 
is  the  distance  from  the  projection  of  the  point 


26 


DESCRIPTIVE  GEOMETRY 


to  tlie  axis,  and  the  other  leg  of  Avhich  is  equal 
to  the  distance  from  the  second  projection  of 
the  point  to  the  ground  line. 

Let  a,  Fig.  69,  be  the  point  in  space  to  be 
revolved  into  H  about  I)  D^  lying  in  H  as  an  . 
axis.     If  a  point  be  revolved  about  an  axis, 
its  locus  will  be  in  a  plane  perpendicular  to 
the  axis.     X  is  this  plane,  and  BX,  its  hori- 


zontal trace,  is  perpendicular  to  D*.  Point  a, 
in  revolving,  will  describe  a  circle  with  ah  as 
a  radius.  The  points  a'  and  a",  in  which  this 
circle  pierces  S,  are  the  required  revolved 
positions  of  the  point  a.  Since  angle  aa''b  is 
a  right  angle,  ha  is  equal  to  the  hypotenuse  of 
a  right  triangle,  one  leg  of  which  is  a''b,  the 
distance  from  a'^  to  the  axis,  and  the  other  leg 


Fig.  71. 


Fi^.  72. 


Tffrfc 


LENGTH   OF   A   LINE 


o? 


is  a*«,  or  the  distance  from  a*  to  (rZ. 

C<»NSTRUCTiON.  Fig.  70.  The  point  a  is 
represented  by  its  two  projections,  a*"  and  a*. 
Through  (^-  draw  ^X  perpendicular  to  7>*. 
The  revolved  position,  a'  or  a",  will  lie  in  HX 
at  a  distance  from  D''  equal  to  the  hypotenuse 
of  a  right  triangle,  one  leg  of  which  is  the  dis- 
tance from  a*  to  the  axis  D*,  and  the  other  leg 
is  the  distance  from  a*  to  GL. 

If  the  axis  lies  in  T',  the  revolved  position 
of  the  point  will  lie  in  a  line  passing  through 
the  vertical  projection  of  the  point  perpen- 
dicular to  the  axis  and  at  a  distance  from  this 
axis  equal  to  the  hypotenuse  of  a  right  tri- 
angle, one  leg  of  which  is  the  distance  from 
the  vertical  projection  of  the  point  to  the  axis, 
while  the  other  leg  is  the  distance  from  the 
•horizontal  projection  of  the  point  to  the  ground 
line. 

39.    To  determine  the  true  length  of  a  line. 

Principle.  A  line  is  seen  in  its  true  length 
on  that  coiirdinate  plane  to  which  it  is  parallel, 
or  in  which  it  lies  (Figs.  16  to  21,  page  9). 

Method.     Revolve  the  line  parallel  to,  or 


into  either  coordinate  plane,  at  which  time  one 
of  its  projections  will  be  parallel  to  the  ground 
line  and  the  other  projection  will  measure  the 
true  length  of  the  line  in  space. 

Case  1.  When  the  line  is  revolved  paral- 
lel to  a  coordinate  plane. 

Construction.  Fig.  71.  Let  ah  he  a.  line 
ill  the  first  quadrant  inclined  to  both  T'^and 
H.  Neither  projection  will  equal  the  true 
length  of  the  line  in  space.  Revolve  ah  about 
the  projecting  line  aa*.  as  an  axis  until  it  is 
parallel  to  V.  Point  a  will  not  move  ;  hence, 
its  projections,  a"  and  a*,  will  remain  station- 
ary. Point  h  will  revolve  in  a  plane  parallel 
to  H^  to  ftj ;  hence,  J*  will  describe  the  arc  6*ftj*, 
and  a^Jj*  will  be  parallel  to  CrL.  Then  6''  will 
move  parallel  to  GL  to  its  new  position  5*,  and 
a'ftJ^  the  new  vertical  projection,  will  equal  the 
true  length  of  line  ah. 

Fig.  72  is  the  orthographic  projection  of  the 
problem.  It  is  of  interest  to  note  that  the 
angle  which  this  true  length  makes  with  GL 
is  the  true  size  of  the  angle  which  the  line  in 
space  makes  with  H. 


28 


DESCRIPTIVE  GEOMETRY 


Figs.  73  and  74  represent  the  line  ah  re- 
volved parallel  to  JjT,  thus  obtaining  the  same 
result  as  to  the  length,  of  line.  Here  the 
angle  which  the  true  length  of  the  line  makes 
with  QL  is  the  true  size  of  the  angle  which 
the  line  in  space  makes  with  V. 

40.  Case  2.  When  tlie  line  is  revolved 
into  a  coordinate  plane. 

Construction.  Figs.  75  and  76  represent 
the  line  ah  of  the  previous  figures  revolved 
into  S  about  its  horizontal  projection  a^h^  as 
an  axis.  Point  a  revolves  in  a  plane  perpen- 
dicular to  the  axis  a*6*  (Art.  38,  page  25); 
therefore,  its  revolved  position,  a\  will  lie  in  a 
line  perpendicular  to  a*J*  and  at  a  distance 
from  a^  equal  to  aa"*^  or  the  distance  from  a"  to 
aL*     Similarly  h'  is  located. 

41.  From  Fig.  76  it  will  be  seen  that  if  the 
revolved  position  of  the  line,  a'h\  be  continued, 
it  will  pass  through  the  point  where  a6,  con- 
tinued, intersects  its  own  projection,  which  is 

*  This  does  not  contradict  Art.  38,  page  26,  in  that  one 
leg  of  the  triangle  is  equal  to  zero. 


the  horizontal  trace  of  the  line,  and  since  it  is 
in  the  axis,  it  does  not  move  in  the  revolution. 
Thus,  in  every  case  where  a  line  is  revolved 
into  one  of  the  coordinate  planes,  its  revolved 
position  will  pass  through  the  trace  of  the  line. 
Figs.  77  and  78  illustrate  a  line  cd  piercing  V 
in  the  point  e.  In  order  to  determine  its  true 
length  it  has  been  revolved  into  K  about  d^d'' 
as  an  axis.  Since  the  point  e  lies  in  the  axis 
and  does  not  move,  point  e  revolves  in  one 
direction  while  point  d  must  revolve  in  the 
opposite  direction,  thus  causing  the  revolved 
position  to  pass  through  the  vertical  trace  of 
the  line. 

Again  it  is  of  interest  to  note  that  when  a 
line  is  revolved  into  a  coordinate  plane  to  de- 
termine its  true  length,  the  angle  whicli  the 
revolved  position  makes  with  the  axis  is  the 
true  size  of  the  angle  which  the  line  in  space 
makes  with  the  coordinate  plane  into  which  it 
has  been  revolved.  Thus  in  Figs.  77  and  78, 
angle  c'e'^&'  is  the  true  size  of  the  angle  which 
line  ctT  makes  with  T''. 


TRUE  LENGTH  OF  A   LINE 


29 


Fig.  76. 


Fig.  77. 


Fig.  78. 


30 


DESCRIPTIVE   GEOMETRY 


42.  Given  a  point  lying  on  a  plane,  to 
determine  its  position  when  the  plane  shall 
have  been  revolved  about  either  of  its  traces 
as  an  axis  to  coincide  with  a  coordinate  plane. 

Fkinciple.  .  This  is  identical  with  the  prin- 
ciple of  Art.  38,  page  25,  since  eitlier  trace  of 
the  plane  is  an  axis  lying  in  a  coordinate  plane, 
one  projection  of  which  is  the  line  itself,  and 
the  other  projection  of  which  is  in  the  ground 
line. 

Method.     See  Art.  38,  page  25. 

Construction.  Fig.  79.  ^iVand  FiV^are 
the  traces  of  the  given  plane  and  6*  and  6",  the 
projections  of  the  point.  If  the  plane  with 
point  h  thereon  be  revolved  into  H  about  HN 
as  an  axis,  the  point  will  move  in  a  plane  per- 
pendicular to  fl!ZV,  and  will  lie  somewhere  in 
V*h'.  Its  position  in  this  line  must  be  deter- 
mined by  finding  the  true  distance  of  the  point 
from  HN.  This  distance  will  equal  the  hy- 
potenuse of  a  right  triangle  of  which  V*d^ 
the  horizontal  projection  of  the  hypotenuse, 
is  one  leg,  and  5''c,  the  distance  of  the  point 
from  H^  is  the  other  leg.     By  laying  off  V'e 


/ 

1 
1 
1 
\ 

J^ 

\ 

e 

"^ 

Fig.  79       W  __- 


ANGLE  BETWEEN  LINES 


31 


eqyal  to  cJ^",  and  perpendicular  to  6*t7,  the  lengtli 
required,  de,  is  determined,  and  when  laid  off 
on  db'  from  rf,  will  locate  the  revolved  position 
of  point  b. 

If  it  were  required  to  revolve  the  point  into 
F"  about  or  as  an  axis,  it  would  lie  at  b"  in  a 
perpendicular  to  T'jV  through  6^  The  dis- 
tance kb"  will  equal  kf,  the  hypotenuse  of  a 
right  triangle,  one  leg  of  which  is  kb",  and  the 
other  leg  of  wliich  is  e(|nal  to  ci*. 

43.  The  revolved  position  of  a  line  lying 
in  a  plane  is  determined  b}"^  finding  the  re- 
volved position  of  two  points  of  the  line.  If 
the  line  of  the  plane  is  parallel  to  the  trace 
which  is  used  as  an  axis,  its  revolved  position 
will  also  be  parallel  to  the  trace. 

If  a  line  has  its  trace  in  the  axis,  this  trace 
will  not  move  during  the  revolution ;  there- 
fore, it  will  be  necessary  to  determine  but  one 
other  point  in  the  revolved  position. 

Fig.  80  represents  the  lines  C  and  D  of  the 
plane  M  revolved  into  JS" about  JIR  as  an  axis. 
The  revolved  position  of  point  a  is  determined 


at  a\  through  which  C  is  drawn  parallel  to 
IIR  (since  line  C  of  the  plane  is  parallel  to 
HR),  and  jy  through  «*,  the  horizontal  trace 
of  line  D. 

44.  To  determine  the  angle  between  two 
intersecting  lines. 

Pkinciple.  The  angle  between  intersect- 
ing lines  maybe  measured  when  the  plane  of  the 
lines  is  revolved  to  coincide  with  one  of  the 
coordinate  planes. 

Method.  1.  Pass  a  plane  through  the  two 
intersecting  lines  and  determine  its  traces 
(Art.  30,  page  20).  2.  Revolve  the  plane 
with  the  lines  thereon,  about  either  of  its 
traces  as  an  axis,  until  it  coincides  with  a  co- 
ordinate plane  (Art.  43,  page  31).  As  it  is 
necessary  to  determine  the  revolved  position 
of  but  one  point  in  each  line,  let  the  point  be 
one  common  to  both  lines,  their  point  of  inter- 
section, and,  therefore,  the  vertex  of  the  angle 
between  tliem.  3.  The  angle  between  the  re- 
volved position  of  the  lines  is  the  required 
angle. 


32 


DESCRIPTIVE  GEOMETRY' 


45.  To  draw  the  projections  of  any  polygon 
having  a  definite  shape  and  size  and  occupying 
a  definite  position  upon  a  given  plane. 

Principle.  The  polygon  will  appear  in  its 
true  size  and  shape,  and  in  its  true  position  on 
the  plane,  when  the  plane  has  been  revolved 
about  one  of  its  traces  as  an  axis  to  coincide 
with  a  coordinate  plane. 

Method.  1.  Revolve  the  given  plane  into 
one  of  the  coordinate  planes  about  its  trace  as 
an  axis.  2.  Construct  the  revolved  position  of 
the  polygon  in  its  true  size  and  shape,  and 
occupying  its  correct  position  on  the  plane. 
•S.  Counter-revolve  the  plane,  with  the  poly- 
gon thereon,  to  its  original  position,  thus 
obtaining  the  projections  of  the  polygon. 

Construction.  Fig.  81.  Let  it  be  re- 
quired to  determine  the  projections  of  a  regu- 
lar pentagon  on  an  oblique  plane  iV,  the  center 
of  the  pentagon  to  be  at  a  distance  x  from  H^ 
and  y  from  V,  and  one  side  of  the  pentagon  to 
be  parallel  to  F",  and  of  a  length  equal  to  z. 

Determine  the  projections  of  the  center  as 
at  0"   and  0''  (Art.   37,  page  24).      Revolve 


plane  N  about  one  of  its  traces  as  an  axis 
until  it  coincides  with  one  of  the  coordinate 
planes  (in  this  case  about  VN  until  it  coin- 
cides with  F'),  0'  being  the  revolved  position 
of  the  center  (Art.  42,  page  30).  About 
0'  as  a  center  draw  the  pentagon  in  its  true 
size  and  shape,  having  one  side  parallel  to  VN, 
and  of  a  length  equal  to  z.  Counter-revolve 
the  plane  to  obtain  the  projections  of  the 
pentagon. 

46.  Counter-revolution.  The  counter- 
revolution may  be  accomplished  in  several 
ways,  of  which  three  are  here  shown. 

Construction  1.  Through  0',  Fig.  81, 
draw  any  line  C  to  intersect  VN  in  w",  and 
connect  if  with  0"  to  obtain  C".  Through  d' 
draw  B'  parallel  to  C  to  intersect  VN  in  wi", 
from  which  point  draw  D"  parallel  to  C. 
From  d'  project  perpendicularly  to  VN  to  in- 
tersect D"  in  c?",  the  vertical  projection  of  one 
point  in  the  required  vertical  projection  of  the 
pentagon.  Similarly  determine  the  vertical 
projections  of  the  other  points  by  drawing 
lines  parallel  to  O. 


PROJECTIONS  OF  A   POLYGON 


33 


47.  Construction  2.  A  sometimes  shorter 
method  of  counter-revolution,  to  obtain  the 
vertical  projection  of  the  pentagon,  is  as 
follows.  Fig.  82.  Assume  the  revolved 
position  of  the  pentagon  to  have  been  drawn 
as  described  above.  From  any  point  of  tlie 
pentagon,  as  c',  draw  a  line  through  the  center 
o\  and  continue  it  to  meet  the  axis  of  revolu- 
tion in  m'.  Then  m^o'  is  the  vertical  projec- 
tion of  this  line  after  counter-revolution  (Art. 
43,  page  31).  and  c'  will  lie  on  nfo'  at  its  inter- 


section with  a  perpendicular  to  \1^  from  c' 
(Art.  42,  page  30).  Produce  c'h'  to  VN,  thus 
determining  the  line  c'6^  and,  therefore,  J". 
Since  h'a'  is  parallel  to  VN^  a'  may  be  found 
by  drawing  a  parallel  to  VN  from  h^  to  meet 
the  perpendicular  from  a!  (Art.  85,  page 
22).  Likewise  d^  is  determined,  since  c'd'  is 
parallel  to  VN.  Next  continue  e'd'  to  the 
axis  at  n";  draw  n'd'  to  meet  o'o'  produced  at 
e".  The  horizontal  projection  of  the  pentagon 
is  best  determined  by  Art.  35,  page  22. 


Fig.  81, 


V     Fig.  82. 


34 


DESCRIPTIVE   GEOMETRY 


48.  Construction  3.  Fig.  83.  Let  it  be 
assumed  that  the  plane  iV"  has  been  revolved 
into  V  about  VN  as  an  axis,  and  that  the 
revolved  position  of  ITN^  has  been  determined, 
as  UN',  by  revolving  any  point  g  of  the  hori- 
zontal trace  of  the  plane  about  FTVas  an  axis 
(Art.  43,  page  31).  The  angle  between  UN' 
and  FiV  is  the  true  size  of  tlie  angle  between 
the  traces  of  the  plane ;  also  the  area  between 
UN '  and  VN  is  the  true  size  of  that  portion 
of  plane  iV  lying  between  its  traces.  Next  let 
it  be  assumed  that  the  polygon  has  been 
drawn  in  its  revolved  position,  and  occupying 
its  correct  location  with  respect  to  VJV  and 
UN,  e'  and  d'  being  two  of  its  vertices..  To 
coiinter-revolve,  continue  e'd'  to  intersect  UN' 
in  k',  and  VJV  in  n".  Since  A:  is  a  point  in  the 
horizontal  trace  of  the  plane,  its  vertical  pro- 
jection will  lie  in  GrL.  Then  k^  will  lie  in 
CrL  at  the  intersection  of  a  perpendicular  to 
VNirom  k',  and  k'^  is  in  SN.  Also  the  inter- 
section of  e'd'  with  the  axis  VWm  w",  and  its 
horizontal  projection  Avill  lie  in  GL  at  w*. 
Draw  w"^",  producing  it  to  intersect  perpen- 


diculars to  in^  from  d'  and  e',  in  d"  and  e". 
Draw  n''k'',  producing  it  to  intersect  perpen- 
diculars to  GrL  from  c?"  and  e",  at  <7^  and  e*. 
Similarly  determine  the  jDrojections  of  the 
other  points  of  the  pentagon. 

A  fourth  construction   for   counter-revolu- 
tion is  explained  in  Art.  91,  page  63. 


UNIVERSIT 

OF 

^*^^^^£SS%NTERSECTION   BETWEEN  PLANES 


35 


49.  To  determine  the  projections  of  the  line 
of  intersection  between  two  planes. 

Principle.  The  line  of  iutersection  be- 
tween two  planes  is  common  to  each  plane; 
therefore,  the  traces  of  this  line  must  lie  in 
the  traces  of  each  plane.  Hence,  the  point  of 
intersection  of  the  vertical  traces  of  the  planes 
is  tbe  vertical  trace  of  the  required  line  of 
intersection,  and  the  point  of  intersection  of 
the  horizontal  traces  of  the  planes  is  the  hori- 
zontal trace  of  the  required  line  of  intersec- 
tion between  the  planes. 

There  may  be  three  cases  as  follows  : 

Case  1.  When  no  auxiliary  plane  is  re- 
quired. 

Case  2.  When  an  auxiliary  plane  parallel 
to  F'or  J3'is  used. 

Case  3.  When  an  auxiliary  plane  parallel 
to  P  is  required. 

50.  Case  1.  When  like  traces  .of  the 
given  planes  may  be  made  to  intersect,  no 
auxiliary  plane  is  required  for  the  solution  of 
the  problem. 

Method.    1.     Determine  the  points  of  in- 


tersection of  like  traces  of  the  planes,  which 
points  are  the  traces  of  the  line  of  intersection 
between  the  planes.  2.  Draw  the  projectfons 
of  the  line  (Art.  26,  page  18). 

Construction.  Figs.  84  and  85  illustrate 
the  principle  and  need  no  explanation,  line  ch 
being  the  line  of  intersection  between  the 
given  planes  N  and  S. 


Fig.  84. 


Fig.  85. 


36 


DESCRIPTIVE  GEOMETRY 


Figs.  86  and  87  illustratie  a  special  condi- 
tion of  Case  1,  in  which  one  of  the  phines  is 
parallel  to  a  coordinate  ^plane.  Plane  iV  is 
inclined  to  both  Fand  H,  and  plane  R  is  par- 
allel to  H.  Since  plane  R  is  perpendicular  to 
F",  its  vertical  trace  will  be  the  vertical  pro- 
jection, e^",  of  the  required  line  of  intersec- 
tion between  planes  N  and  R.  As  e^  is  the 
vertical  trace  of  line  ef,  the  horizontal  projec- 
tion of  this  line  will  be  g'/*,  which  is  parallel 
to  HN  (Art.  35,  page  22). 

51.  Case  2.  When  one  or  both  pairs  of 
like  traces  of  the  given  planes  do  not  intersect 
within  the  limits  of  the  drawing,  and  are  not 
parallel,  or  when  all  the  traces  meet  the 
ground  line  in  the  same  point,  auxiliary  cut- 
ting planes  parallel  to  either  V  or  IT  may  be 
used  for  the  solution  of  the  problem. 

Method.  1.  Pass  an  auxiliary  cutting 
plane  parallel  to  V  or  JT,  and  determine  the 
lines  of  intersection  between  the  auxiliary 
plane  and  each  of  the  given  planes.  Their 
point  of  intersection  will  be  common  to  the 
given  planes,  and,  therefore,  a  point  in  the  re- 
% 


quired  line.  2.  Determine  a  second  point  by 
passing  another  auxiliary  plane  parallel  to  V 
or  H,  and  the  required  line  will  be  determined. 

Construction.  Fig.  88.  represents  two 
planes,  T  and  M,  with  their  horizontal  traces 
intersecting,  but  their  vertical  traces  inter- 
secting beyond  the  limits  of  the  drawing. 

By  Case  1,  point  d  is  one  point  in  the  line 
of  intersection  between  the  planes.  To  obtain 
a  second  point  an  auxiliary  plane  X  has  been 
passed  parallel  to  V.  Then  JIX  is  parallel  to 
GrL,  and  C"^,  the  horizontal  projection  of  the 
line  of  intersection  between  planes  X  and  M, 
lies  in  IIX,  while  (7"  is  parallel  to  VM  (Case 
1).  Likewise  J5*,  the  horizontal  projection  of 
the  line  of  intersection  between  planes  X  and 
T,  lies  in  ffX,  while  B"  is  parallel  to  VT. 
Then  point  a,  the  point  of  intersection  between 
lines  B  and  (7,  is  a  point  in  the  line  of  inter- 
section between  planes  i^f  and  T^  since  point 
a  lies  in  line  0  of  plane  iHf,  and  in  line  B  of 
plane  T.  Line  ad  is,  therefore,  the  required 
line  of  intersection  between  the  two  given 
planes. 


INTERSECTION  BETWEEN  PLANES 


37 


Fig.  86. 


Fig.  87. 


Fig.  89. 


If  neither  the  vertical  nor  horizontal  traces 
of  the  given  planes  intersect  within  the  limits 
of  the  drawing,  it  will  be  necessary  to  use  two 
auxiliary  cutting  planes,  both  of  which  may  be 
parallel  to  F",  both  parallel  to  IT,  or  one  parallel 
to  F'and  one  parallel  to  H. 

"52.  If  two  traces  of  the  given  planes  are 
parallel,  the  line  of  intersection  between  them 
will  be  parallel  to  a  coordinate  plane,  and  will 
have  one  projection  parallel  to  the  ground  line 
and  the  other  projection  parallel  to  the  parallel 
traces  of  the  given  planes. 

53.  Fig.  89  represents  a  condition  when  all 
four  traces  of  the  given  planes  intersect  GL 
in  the  same  point,  neither  plane  containing 
GrL.  This  point  of  intersection  of  the  traces, 
i,  is  one  point  in  the  required  line  of  inter- 
section between  the  planes  (Case  1).  A 
second  point,  a,  is  obtained  by  passing  the 
auxiliary  cutting  plane  X  parallel  to  H^  inter- 
secting plane  N  in  line  0  and  plane  aS'  in  line 
D.  Point  a,  the  point  of  intersection  of  these 
lines,  is  a  second  point  in  the  required  line  of 
intersection,  ah,  between  the  given  planes  N 
and  S  (Case  2).  * 


38 


DESCRIPTIVE  GEOMETRY 


54.  Case  3.  When  both  intersecting 
planes  are  parallel  to  the  ground  line,  or 
when  one  of  the  intersecting  planes  contains 
the  ground  line. 

Method.  1.  Pass  an  auxiliary  cutting  plane 
parallel  to  P.  2.  Determine  its  line  of  inter- 
section with  each  of  the  given  planes. 
3.  The  point  of  intersection  of  these  two  lines 
is  one  point  in  the  required  line  of  intersection 
between  the  two  given  planes.  It  is  not  nec- 
essary to  determine  a  second  point,  for,  when 
both  given  planes  are  parallel  to  the  ground 
line,  their  line  of  intersection  will  be  parallel 
to  the  ground  line. 

When  one  plane  contains  the  ground  line, 
one  point  in  the  line  of  intersection  is  the  point 
of  intersection  of  all  the  traces.    (Figs.  91,  92.) 

Construction.  In  Fig.  90  the  two  planes 
iVand  /S'are  parallel  to  CrL.  Pass  the  auxil- 
iary profile  plane  P  intersecting  plane  N  in 
the  line  whose  profile  projection  is  i>^,  and  the 
plane  S  in  the  line  whose  profile  projection  is 
B^.  A^,  the  intersection  of  these  two  lines,  is 
the  profile  projection  of  one  point  in  the  line 


of  intersection  between  planes  iV  and  S ',  in 
fact  A^  is  the  profile  projection  of  the  re- 
quired line,  and  A*  and  A'^  are  the  required 
projections. 

55.  In  Fig.  91  the  plane  S  contains  GL  and, 
therefore,  it  cannot  be  definitely  located  with- 
out its  profile  trace,  or  its  angle  with  a  coordi- 
nate plane,  and  the  quadrants  through  which 
it  passes.  Plane  iVis  inclined  to  V,II,  and  P, 
and  plane  S  contains  (ri,  passing  through  1 Q 
and  3Q  at  an  angle  6  with  V.  As  in  the  pre- 
vious example  the  profile  auxiliary  plane  is 
required  and  PiV,  the  profile  trace  of  N,  is 
determined  as  before.  PS,  the  profile  trace 
of  S,  is  next  drawn  through  1 Q  and  3  Q  and 
making  an  angle  6  with  VP.  Then  d^,  the 
intersection  of  PiV and  PS,  is  the  profile  projec- 
tion of  one  point  in  the  required  line  of  inter- 
section between  planes  N  and  S,  and  d^  and 
d'^  are  the  required^  projections  of  this  point. 
The  horizontal  and  vertical  traces  of  this  line 
of  intersection  are  at  5,  for  it  is  here  that  VN 
and  VS  intersect,  and  also  where  ^iVand  ^aS' 
intersect.     The  projections  of  two  points  of 


INTERSECTION   BETWEEN  PLANES 


39 


the  line  having  now  been  determined,  the  pro- 
jections of  the  line,  dh,  may  be  drawn. 

Case  3  is  apj^licable  to  all  forms  of  this  prob- 
lem, and  Fig.  92  represents  the  planes  N  and 
aS'  of  Fig.  89  with  their  line  of  intersection,  ah, 
determined  by  this  method. 


HS 


.^-                    1^5        / 

k 

/ 

Fig.  90. 

[Fig.  92. 


40 


DESCRIPTIVE   GEOMETRY 


56.  Revolution,  Quadrants,  and  Counter  rev- 
olution. Fig.  93.  Let  it^  be  required  to  pass 
a  plane  >S' through  the  iirst  and  third  quadrants, 
making  an  angle  0  with  F'and  intersecting  the 
oblique  plane  iVin  the  line  A.  The  problem  is 
solved  by  Art.  5-1,  page  38,  but  this  question 
may  arise:  In  what  direction  shall  Pas' be  drawn 
and  with  what  line  shall  it  make  the  given 
angle  6?  Conceive  the  auxiliary  profile  plane 
P  to  be  revolved  about  VP  as  an  axis  until  it 
coincides  with  F,  that  portion  of  P  which  was 
in  front  of  V  to  be  revolved  to  the  right. 
Then  the  line  VP  HP  represents  the  line  of 
intersection  between  VnndP;  (ri  represents 
the  line  of  intersection  between  H  and  P;  and 
PN,  the  line  of  intersection  between  iVand  P. 
That  portion  of  the  paper  lying  above  GrL  and 
to  the  right  of  VP  will  represent  that  portion 
of  P  which,  before  revolution,  was  in  IQ; 
above  (rX  and  to  the  left  of  VP,  in  2Q;  below 
GL  and  to  the  left  of  VP,  in  3Q;  and  below 
GL  and  to  the  right  of  VP,  in  4Q.  Since  jS 
is  to  pass  through  IQ  and  3Q,  making  an  angle 
6  with  v.,  the  line  PS,  in  which  S  intersects 


P,  should  be  drawn  on  that  portion  of  P  which 
is  in  IQ  and  3Q,  and  should  pass  through  the 
point  of  intersection  of  VP  and  GL,  making 
the  angle  6  with  VP.  P/S  iind  PiV^  intersect  in 
d^,  the  profile  projection  of  one  point  in  the 
required  line  of  intersection.  In  counter- 
revolution, that  is,  revolving  P  back  into  its 
former  position  perpendicular  to  both  Fand  H, 
rotation  will  take  place  in  a  direction  opposite 
to  that  of  the  first  revolution,  and  d"  and  d'^ 
will  be  as  indicated. 

Fig.  94  represents  the  same  problem  when 
the  auxiliary  profile  plane  P  has  been  revolved 
in  the  opposite  direction  to  coincide  with  F. 

Figs.  95  and  96  are  examples  of  the  same 
problem  when  the  auxiliary  profile  plane  P 
has  been  revolved  about  IIP  as  an  axis  until  it 
coincides  witli  II.  Then  GL  will  represent 
the  revolved  position  of  the  line  of  intersection 
between  F  and  P,  and  the  line  VP  HP  will 
represent  the  revolved  position  of  the  line  of 
intersection  between  ^and  P.  PS  will  then 
make  its  angle  B  with  GL,  and  its  direction 
will  be  governed  by  the  rotation  assumed. 


OOU  NTER  -  REVOLUTION 


41 


Fig.  95 


2 

Q 

V 

k 
^^/^' 

y 

\ 

6*6^^ 

""i  1 
N  1 

y^  vs 

HS     \ 

-^4^ 

^ 

rf* 

A^ 



1Q 

ft. 
% 

\^ 

\ 

4Q 

Fig.  96. 


42 


DESCRIPTIVE  GEOMETRY 


57.  To  determine  the  point  in  which  a  line 
pierces  a  plane. 

Method.  1.  Pass  a^ii  auxiliary  plane 
through  the  line  to  intersect  the  given  plane. 
2.  Determine  the  line  of  intersection  between 
the  given  and  auxiliary  planes.  3.  The  re- 
quired point  will  lie  at  the  intersection  of  the 
given  line  and  the  line  of  intersection  between 
the  given  and  auxiliary  planes. 

There  may  be  four  cases  as  follows: 

Case  1.  When  any  auxiliary  plane  contain- 
ing the  line  is  used. 

Case  2.  When  the  horizontal  or  vertical 
projecting  plane  of  the  line  is  used. 

Case  3.  When  the  given  line  is  parallel  to 
P,  thus  necessitating  the  use  of  an  auxiliary 
profile  plane  containing  the  line. 

Case  4.  When  the  given  plane  is  defined  by 
two  lines  which  are  not  the  traces  of  the  plane. 

58.  Case  1.  When  any  auxiliary  plane  con- 
taining the  line  is  used. 

Construction.  Fig.  97.  Let  A  be  the 
given  line  and  iV  the  given  plane.  Through 
line  A  pass  any  auxiliary  plane  Z  ( Art.  29,  page 


20)  intersecting  plane  iV  in  line  C  (Art.  50, 
page  35).  Since  lines  A  and  die  in  plane  Z,  d 
is  their  point  of  intersection,  and  since  C  is  a 
line  of  plane  iV,  point,  ci  is  common  to  both  line 
A  and  plane  N;  hence,  their  intersection. 

59.  Case  2.  When  the  horizontal  or  verti- 
cal projecting  plane  of  the  line  is  used. 

Construction.  Fig.  98  represents  the 
same  line  A  and  plane  iVof  the  previous  figure. 
Pass  the  horizontal  projecting  plane  X  of  line 
A  (Art.  8,  page  7),  intersecting  plane  iV  in 
line  C  (Art.  50,  page  35).  Lines  A  and  C  in- 
tersect in  point  d,  the  required  point  of  pierc- 
ing of  line  A  and  plane  N". 

Fig.  99  is  the  solution  of  the  same  problem 
by  the  use  of  plane  Y,  the  vertical  projecting 
plane  of  line  A. 

60.  Case  3.  When  the  given  line  is  parallel 
to  P,  thus  necessitating  the  use  of  an  auxiliary 
profile  plane  containing  the  line. 

Construction.  Fig.  100.  Let  N  be  the 
given  plane  and  line  ab,  parallel  to  P,  the  given 
line.  Pass  an  auxiliary  profile  plane  P  through 
the  given  line  ah,  intersecting  plane  iVin  the 


PIERCING  OF  THE  LINE  AND  PLANE 


43 


line   (7,  liaving    C"  for   its  profile    projection  profile    projection   of   the   required   point   of 

(Art.  54,  page  38).     Also  determine  a^J^,  the  piercing  of  line  oA  and  plane  N.     The  vertical 

profile  projection  of  the  given  line  (Art.  21,  and   horizontal  projections  of  this   point   are 

page  14).      C-P  and  a^hp  intersect  in  dp^  the  £?"  and  rf*. 


Fig.  99 


44 


DESCRIPTIVE   GEOMETRY 


6i.  Case  4.  When  the  given  plane  is  de- 
fined by  two  lines  which  are  not  its  traces.    . 

Construction.  Figs.  101  and  102.  Let 
the  given  plane  be  defined  by  lines  B  and  2), 
and  let  A  be  the  given  line  intersecting  this 
plane  at  some  point  to  be  determined.  This 
case  should  be  solved  without  the  use  of  the 
traces  of  any  plane.  Pass  the  horizontal  pro- 
jecting plane  of  line  A  intersecting  line  B  in 
point  e,  and  line  D  in  point/,  and  consequently 
intersecting  the  plane  of  lines  B  and  D  in  line 
ef.  Because  this  horizontal  projecting  plane 
of  line  A  is  perpendicular  to  H  all  lines  lying 
in  it  will  have  their  horizontal  projections 
coinciding,  and  e^f^  will  be  the  horizontal  pro- 
jection of  the  line  of  intersection  between  the 
auxiliary  plane  and  the  plane  of  lines  B  and 
D.  Draw  the  vertical  projection  of  this  line 
through  the  vertical  projections  of  points  e  and 
/;  its  intersection  with  vl*',  at  c?",  will  be  the 
vertical  projection  of  the  required  point  of 
piercing  of  line  A  with  the  plane  of  lines  B 
and  B. 

The  same  result  may  be  obtained  by  the  use 


of  the  vertical  projecting  plane  of  line  A. 

62.  If  a  right  line  is  perpendicular  to  a  plane, 
the  projections  of  that  line  will  be  perpendicular 
.to  the  traces  of  the  plane.     In  Figs.  103  and 

104  HN  and  VN  are  the  traces  of  a  plane  to 
which  line  A  is  perpendicular.  The  horizontal 
projecting  plane  of  line  A  is  perpendicular  to 
jET. by  construction;  it  is  also  perpendicular  to 
plane  N  because  it  contains  a  line,  A,  perpen- 
dicular to  iV;  therefore,  being  perpendicular 
to  two  planes  it  is  perpendicular  to  their  line 
of  intersection,  HN.  But  HN'\&  perpendicular 
to  every  line  in  the  horizontal  projecting  plane 
of  line  A  which  intersects  it,  and,  therefore, 
perpendicular  to  A''.     Q.E.D. 

In  like  manner  VN  may  be  proved  to  be 
perpendicular  to  A .  The  converse  of  this 
proposition  is  true. 

63.  To  project  a  point  on  to  an  oblique  plane. 
Principle.     The  projection  of  a  point  on 

any  plane  is  the  intersection  with  that  plane 
of  a  perpendicular  let  fall  from  the  point  to 
the  plane. 

Method.     1.    From  the  given  point  draw 


A  PERPENDICULAR  TO  A  PLANE 


45 


Fig.  i03. 


Fig.  104. 


a  perpendicular  to  the  given  plane  (Art.  62, 
page  44).  2.  Determine  the  point  of  piercing 
of  this  perpendicular  and  the  plane  (Art.  57, 
page  42).  This  point  of  piercing  is  the  re- 
quired projection  of  the  given  point  upon  the 
given  oblique  plane. 

64.  To  project  a  given  line  on  to  a  given 
oblique  plane. 

Method.  If  the  line  is  a  right  line,  project 
two  of  its  points  (Art.  63,  page  44).  If  the 
line  is  curved,  project  a  sufficient  number  of 
its  points  to  describe  the  curve. 

65.  To  determine  the  shortest  distance  from 
a  point  to  a  plane. 

Pkinciple.  The  shortest  distance  from  a 
point  to  a  plane  is  the  perpendicular  distance 
from  the  point  to  the  plane. 

Method.  1.  From  the  given  point  draw  a 
perpendicular  to  the  given  plane  (Art.  62, 
page  44).  2.  Determine  the  point  of  piercing 
of  the  perpendicular  and  the  plane  (Art.  57, 
page  42).  3.  Determine  the  true  length  of 
the  perpendicular  between  the  point  and  the 
plane  (Art.  39,  page  27). 


46 


DESCRIPTIVE  GEOMETRY 


66.  Shades  and  Shadows.  The  graphic 
representation  of  objects,  especially  those  of 
an  architectural  character,  may  be  made  more 
effective  and  more  easily  understood  by  draw- 
ing the  shadow  cast  by  the  object. 

When  a  body  is  subjected  to  rays  of  light, 
that  portion  which  is  turned  away  from  the 
source  of  light,  and  which,  therefore,  does  not 
receive  any  of  its  rays,  is  said  to  be  in  shade. 
See  Fig.  105.  When  a  surface  is  in  light  and 
an  object  is  placed  between  it  and  the  source 
of  light,  intercepting  thereby  some  of  the  rays, 
that  portion  of  the  surface  from  which  light  is 
thus  excluded  is  said  to  be  in  shadow.  That 
portion  of  space  from  which  light  is  excluded 
is  called  the  umbra  or  invisible  shadow. 

(a)  The  umbra  of  a  point  in  space  is  evi- 
dently a  line. 

(6)  The  umbra  of  a  line  is  in  general  a 
plane. 

(c?)  The  umbra  of  a  plane  is  in  general  a 
solid. 

(c?)  It  is  also  evident  from  Fig.  105  that 
the  shadow  of  an  object  upon  another  object 


is  the  intersection  of  the  umbra  ofHhe  first 
object  with  the  surface  of  the  second  object. 

The  line  of  separation  between  the  portion  of 
an  object  in  light  and  the  portion  in  shade  is 
called  the  shade  line.  It  is  evident  from  Fisr. 
105  that  the  shade  line  is  the  boundary  of  the 
shade.  It  is  also  evident  that  the  shadow  of 
the  object  is  the  space  inclosed  by  the  shadow 
of  its  shade  line. 

The  source  of  light  is  supposed  to  be  at  an 
infinite  distance;  therefore,  the  rays  of  light 
will  be  parallel  and  will  be  represented  by 
straight  lines.  The  assumed  direction  of  the 
conventional  ray  of  light  is  that  of  the  diag- 
onal of  a  cube,  sloping  downward,  backward, 
and  to  the  right,  the  cube  being  placed  so  that 
its  faces  are  either  parallel  or  perpendicular  to 
F;  H,  and  P  (Fig.  106).  This  ray  of  light 
makes  an  angle  of  35°  15'  52"  with  the  coordi- 
nate planes  of  projection,  but  from  Figs.  106 
and  107  it  will  be  observed  that  the  projections 
of  the  ray  are  diagonals  of  squares,  and  hence, 
they  make  angles  of  45°  with  GL. 

Since  an  object  is  represented  by  its  projec- 


/ 


SHADES  AND  SHADOWS 


47 


tions,  the  ray  of  light  must  be  represented  by 
its  projections. 

An  object  must  be  situated  in  tlie  first 
quadrant  to  cast  a  shadow  upon  both  F'and  H. 

67.  To  determine  the  shadow  of  a  point  on 
a  given  surface  pass  the  umbra  of  the  point,  or 
as  is  generally  termed,  pass  a  ray  of  light 
through  the  point  and  determine  its  intersec- 
tion with  the  given  surface  by  Art.  24,  page 
16,  or  by  Art.  57,  page  42,  according  as  the 
shadow  is  required  on  a  coordinate  or  on  an 
oblique  plane. 

68.  To  determine  the  shadow  of  a  line  upon 
a  given  surface  it  is  necessary  to  determine 
the  intersection  of  its  umbra  with  that  surface. 
If  the  line  be  a  right  line,  this  is  generally  best 
accomplished  by  finding  the  shadows  of  each 
end  of  the  line  and  joining  them.  If  the  line 
be  curved,  then  the  shadows  of  several  points 
of  the  line  must  be  obtained. 

In  Figs.  108  and  109  cb  is  an  oblique  line 
having  one  extremity,  c,  in  H.  By  passing 
the  ray  of  light  through  h  and  locating  its 
horizontal  trace,  the  shadow  of  h  is  found  to 


fall  upon  H  at  J**.  Since  c  lies  in  J5i  it  is  its 
own  shadow  upon  jGT;  therefore,  the  shadow  of 
line  cb  upon  H  is  c*i**. 


48 


DESCRIPTIVE   GEOMETRY 


69.  To  determine  the  shadow  of  a  solid  upon 
a  given  surface  it  is  necessary  to  determine  the 
shadows  of  its  shade  lines.  When  it  is  diffi- 
cult  to  recognize  which  lines  of  the  object  are 
its  shade  lines,  it  is  well  to  cast  the  shadow  of 
every  line  of  the  object.  The  outline  of  these 
shadows  will  be  the  required  result. 

Fig.  110  represents  an  hexagonal  prism 
located  in  the  first  quadrant  with  its  axis  per- 
pendicular to  H.  Its  shadow  is  represented 
as  falling  wholly  upon  FJ  as  it  would  appear 
if  H  were  removed.  Fig.  Ill  represents  the 
prism  when  the  shadow  falls  wholly  upon  H^ 
as  it  would  appear  if  F'were  removed.  Fig. 
112  represents  the  same  prism  when  both  V 
and  H  are  in  position,  a  portion  of  the  shadow 
falling  upon  F",  and  a  portion  falling  upon  H. 
It  is  now  readily  observed  that  the  shade  lines 
are  he,  ed,  de,  ep,  ps,  sg,  gk,  and  kh,  and  that 
the  shadow  of  the  prism  is  the  polygon  in- 
closed by  the  shadows  of  these  shade  lines. 

When  an  object  is  so  located  that  its 
shadow  falls  partly  upon  V  and  partly  upon 
H^  it  is  generally  best  to  determine  first,  its 


complete  shadow  upon  both  V  and  5",  and  to 
retain  only  such  portions  of  the  shadow  as 
fall  upon  F" above  H,  and  upon  ^before  V. 

Fig.  113  represents  a  right  hexagonal  pyra- 
mid resting  upon  an  oblique  plane  N  and 
having  its  axis  perpendicular  to  that  plane. 
Its  shadow  has  been  cast  upon  iV,  and  the 
construction  necessary  to  determine  the 
shadow  of  one  point,  a,  has  been  shown  (Art. 
59,  page  42). 

From  the  foregoing  figures  these  facts  will 
be  observed : 

If  a  point  lies  on  a  plane,  it  is  its  own  shadow 
upon  that  plane.  See  point  C,  Figs.  108, 109, 
and  the  apex  of  the  pyramid,  Fig.  113. 

If  a  line  is  parallel  to  a  plane,  its  shadow 
upon  that  plane  will  be  parallel,  and  equal  in 
length,  to  the  line  in  space.  Lines  hk  and  ep. 
Fig.  110,  are  parallel  to  V',  lines  he  and  8jo,  cd 
and  QB,  de  and  gk,  Fig.  Ill,  are  parallel  to  H, 
and  the  sides  of  the  hexagon.  Fig.  113,  are 
parallel  to  N. 

If  two  lines  are  parallel,  their  shadows  are 
parallel.      Observe  that  the  shadows  of   the 


SHADES  AND  SHADOWS 


49 


opposite  sides  of  the  liexagons,  and  of  the  lat- 
eral edges  of  the  prisms,  are  parallel. 

If  a  line  is  perpendicular  to  a  coordinate 
plane,  its  shadow  on  that  plane  will  fall  on 
the  projections  of  the  ra^s  of  light  passing 
through  it.  Lines  hk  and  ep.  Fig.  Ill,  fulfill 
this  condition. 


a'd'/'       Ce'd' 


a'  b'f       c'e'  d 


Fig.  113. 
a"  b'  r      c'e'  d' 


50 


DESCRIPTIVE   GEUMETRY 


70.  Through  a  point  or  line  to  pass  a  plane 
having  a  defined  relation  to  a  given  line  or 
plane.    There  may  be  five  cases,  as  follows  : 

Case  1.  To  pass  a  plane  through  a  given 
point  parallel  to  a  given  plane. 

Case  2.  To  pass  a  plane  through  a  given 
point  perpendicular  to  a  given  line. 

Case  3.  To  pass  a  plane  through  a  given 
point  parallel  to  two  given  lines. 

Case  4.  To  pass  a  plane  through  a  given 
line  parallel  to  another  given  line. 

Case  5.  To  pass  a  plane  through  a  given 
line  perpendicular  to  a  given  plane. 

In  cases  1  and  2  the  directions  of  the  re- 
quired traces  are  known.  In  cases  3,  4,  and 
5  two  lines  of  the  required  plane  are  known. 

71.  Casev  I.  To  pass  a  plane  through  a  given 
point  parallel  to  a  given  plane. 

Principle  Since  the  required  plane  is  to 
be  parallel  to  the  given  plane,  their  traces  will 
be  parallel,  and  a  line  through  the  given  point 
parallel  to  either  trace  will  determine  the 
plane. 

Method.      1.    Through    the   given    point 


pass  a  line  parallel  to  one  of  the  coordinate 
planes  and  lyiJig  in  the  required  plane  (Art. 
9,  page  8).  2.  Determine  the  trace  of  this 
line,  thus  determining  one  point  in  the  re- 
quired trace  of  the  plane.  3.  Draw  the 
traces  parallel  to  those  of  the  given  plane. 

Construction.  Fig.  114.  Let  N  be  the 
given  plane  and  b  the  given  point.  Through 
b  pass  line  A  parallel  to  V  and  in  such  a  di- 
rection that  it  will  lie  in  the  required  plane, 
A''  being  parallel  to  GL  and  A"  parallel  to  VN 
(Art.  35,  page  22).  Determine  d,  the  hori- 
zontal trace  of  line  A,  and  through  d'^  draw 
ffS,  the  horizontal  trace  of  the  required  plane, 
parallel  to  HK      VS  will  be  parallel  to  A\ 

72.  Case  2.  To  pass  a  plane  through  a  given 
point  perpendicular  to  a  given  line. 

Principle.  Since  the  required  plane  is  to, 
be  perpendicular  to  the  given  line,  the  traces  of 
the  plane  will  be  perpendicular  to  the  projec- 
tions of  the  line  (Art.  62,  page  44).  Hence, 
a  line  drawn  through  the  given  point  parallel 
to  either  coordinate  plane,  and  lying  in  the 
required  plane,  will  determine  this  plane. 


RELATION  BETWEEN   LINES  AND  PLANES 


51 


Fig.  116. 


Method.  1.  Through  the  given  point  pass 
a  line  parallel  to  one  of  the  coordinate  planes 
and  lying  in  the  required  plane.  2.  Deter- 
mine the  trace  of  this  line,  thus  determining 
one  point  in  the  required  trace  of  the  plane. 
3.  Draw  the  traces  perpendicular  to  the  pro- 
jections of  the  given  line. 

Construction.  Fig.  115.  Through  point 
h  draw  line  C  parallel  to  T^and  lying  in  the 
required  plane  S.  O'  will  be  perpendicular 
to  A'  (Art.  62,  page  44).  RS  and  VS  are 
the  required  traces. 

73.  Case  3.  To  pass  a  plane  through  a  given 
point  parallel  to  two  given  lines. 

Principle.  The  required  plane  will  con- 
tain lines  drawn  through  the  given  point 
parallel  to  the  given  lines. 

Method.  1.  Through  the  given  point  pass 
two  lines  parallel  to  the  two  given  lines. 
2.  Determine  the  plane  of  these  lines  (Art. 
30,  page  20). 

Construction.  Fig.  116.  Given  lines  A 
and  B  and  point  b.  Through  point  b  pass  lines 
C  and  D  parallel  respectively  to  Hues  A  and  B. 
Si  the  plane  of  these  lines,  is  the  required  plane. 


52 


DESCRIPTIVE   GEOMETRY 


74.  Case  4.  To  pass  a  plane  through  a  given 
line  parallel  to  another  given  line. 

Principle.  The  required  plane  will  con- 
tain one  of  the  given  lines  and  a  line  inter- 
secting it  and  parallel  to  the  second. 

Method.  1.  Through  any  ]wint  of  the 
given  line  pass  a  line  parallel  to  the  second 
given  line.  2.  Determine  the  plane  of  these 
intersecting  lines  (Art.  30,  page  20). 

75.  Case  5.  To  pass  a  plane  through  a  given 
line  perpendicular  to  a  given  plane. 

Principle.  The  required  plane  will  con- 
tain the  given  line  and  a  line  intersecting  this 
line  and  |)erpendicular  to  the  given  plane. 

Method.  1.  Through  any  point  of  the 
given  line  pass  a  line  perpendicular  to  the 
given  plane.  2.  Determine  the  plane  of  these 
lines. 

CoNSTRUcrTnifr-^fiig.  117.  Given  line  A 
and  plane  N.  Through  any  point  of  line  A 
pass  line  O  perpendicular  to  plane  iV  (Art. 
62,  page  44).  Determine  S,  the  plane  of  lines 
A  and  C  (Art.  80,  page  20). 

76.  Special  conditions  and  methods  of  Art.  70. 
Case  1.*     To  pass  a  plane  through  a  given 


point  parallel  to  a  given  plane. 

Fig.  118  illustrates  a  condition  in  which  the 
given  plane  iVis  parallel  to  (ri/,  which  neces- 
sitates the  use  of  an  auxiliary  profile  plane. 
Through  the  given  point  b  pass  the  profile 
plane  P.  Determine  b^  and  PiV,  and  tliroug^i 
bP  draw  PS^  the  profile  trace  of  the  required 
plane,  parallel  to  PiV",  whence  VS  and  ^*S',  the 
traces  of  the  required  plane,  are  determined. 

This  condition  may  also  be  solved,  Fig.  119, 
by  passing  a  line  A  through  the  given  point 
i,  parallel  to  ani/  line  C  of  the  given  plane 
iV.  Through  the  traces  of  line  A  the  traces 
of  the  required  plane  *S'  are  drawn  parallel  to 
^iVand  VN,  respectively. 

This  method  is  applicable  to  all  conditions 
of  Case  1. 

77.  Case  2.*  To  pass  a  plane  through  a 
given  point  perpendicular  to  a  given  line. 

Fig.  120  illustrates  a  condition  in  which  the 
projections  of  the  given  line  are  perpendicular 
to  GrL ;  hence,  the  traces  of  the  required  plane 
will  be  parallel  to  GrL.     Let  ac  be  the  given 

*  The  numbers  of  the  cases  are  the  same  as  those  in 
Art.  70,  page  50. 


RELATION  BETWEEN  LINES  AND  PLANES 


53 


vs 

y 

VN 

>  1      ! 

j/\ 

!    ^/ 

\/       HN 

^' 

HS 

b — i'    y 

*     ,-^'       HS 


Fig.  122. 


line  and  h  the  given  point.  Pass  an  auxiliary 
profile  plane  P,  and  determine  a^e^,  the  pro- 
file projection  of  the  given  line,  and  &^,  the 
profile  projection  of  the  given  point.  Through 
h^  draw  PS^  the  profile  trace  of  the  required 
plane,  perpendicular  to  a^e^,  whence  VS  and 
^*S'  are  determined. 

78.  Case  5.*  To  pass  a  plane  through  a 
given  line  perpendicular  to  a  given  plane. 

Fig.  121  illustrates  a  condition  in  which  the 
given  line  ah  is  parallel  to  P.  This  may  be 
solved  by  the  use  of  an  auxiliary  profile  plane, 
or  by  the  following  method:  Through  two 
points  of  the  given  line  ab  pass  auxiliary  lines 
C  and  D  perpendicular  to  the  given  plane  iV, 
and  determine  *S,  the  plane  of  these  parallel 
lines.  Then  S  is  the  required  plane  contain- 
ing line  ah  and  perpendicular  to  plane  N. 

Fig.  122  illustrates  a  condition  in  which  the 
given  plane  N  is  parallel  to  GL.  This  solu- 
tion is  identical  with  that  of  Art.  75,  page  52, 
save  that  to  determine  the  traces  of  the  auxil- 
iary line  B,  an  auxiliary  profile  plane  P  has 
been  used.  Since  ^  is  to  be  perpendicular  to 
N^  Bp  must  be  perpendicular  to  PIf. 


54 


DESCRIPTIVE   GEOMETRY 


79.  To  determine  the  projections  and  true 
length  of  the  line  measuring  the  shortest  distance 
between  two  right  lines  not  in  the  same  plane. 

Principle.  The  shot-test  distance  between 
two  right  lines  not  in  the  same  plane  is  the 
perpendicular  distance  between  them,  and 
only  one  perpendicular  can  be  drawn  termi- 
nating in  these  two  lines. 

Method.  1.  Through  one  of  the  given 
lines  pass  a  plane  parallel  to  the  second  given 
line.  2.  Project  the  second  line  on  to 
the  plane  passed  through  the  first.  3.  At  the 
point  of  intersection  of  the  first  line  and  tlie 
projection  of  the  second  erect  a  perpendicular 
to  the  plane.  This  perpendicular  will  inter- 
sect the  second  line.  4.  Determine  the  true 
length  of  the  perpendicular,  thus  obtaining  the 
required  result. 

Construction.  Figs.  123  and  124.  Given 
lines  A  and  B.  Pass  the  plane  S  through 
line  A  parallel  to  line  B  (Art.  74,  page  52). 
Project  line  B  on  to  plane  S  at  5j,  using  the 
auxiliary  line  I)  perpendicular  to  plane  S  and 
intersecting  it  at  point  k  (Art.  64,  page  45).' 


Since  line  B  is  parallel  to  plane  S,  B^  will  be 
parallel  to  B ;  hence,  their  projections  are 
parallel  (Art.  13,  page  8).  B^  intersects  A 
at  w,  at  which  point  erect  the  required  line  E 
perpendicular  to  plane  S  (Art.  62,  page  44), 
intersecting  line  B  at  0.  Determine  the  true 
length  of  line  E  (not  shown  in  the  figure)  by 
Art.  39,  page  27. 

80.  To  determine  the  angle  between  a  line 
and  a  plane. 

Principle.  The  angle  which  a  line  makes 
with  a  plane  is  the  angle  which  the  line  makes 
with  its  projection  on  the  plane,  or  the  com- 
plement of  the  angle  which  the  line  makes 
with  a  perpendicular  which  may  project  any 
point  of  the  line  on  to  the  plane.  The  line  in 
space,  its  projection  on  the  plane,  and  the 
projector,  form  a  right-angled  triangle. 

Method.  1.  Through  any  point  of  the 
given  line  drop  a  perpendicular  to  the  given 
plane.  2.  Determine  the  angle  between  this 
perpendicular  and  the  given  line.  The  angle 
thus  determined  is  the  complement  of  the 
required  angle. 


ANGLE  BETWEEN  LINE  AND  PLANE 


55 


Construction.  Fig.  125.  Given  line  A 
and  plane  N.  Through  any  point  <r,  of  line  ^4, 
pass  line  B  perpendicular  to  plane  H  (Art. 
62,  page  44).  Determine  one  of  the  traces  of 
the  plane  of  lines  A  a^d  B^  as  VS.  Revolve 
lines  A  and  B  into  F' about  VS  as  an  axis,  as 
at  A'  and  B'  (Art.  43,  page  31).  Then 
angle  d'c'e'  is  the  true  size  of  the  angle  be- 
tween lines  A  and  B,  and  its  complement, 
'e^c'f,  is  the  required  angle  between  line  A  and 
plane  N  (Art.  44,  page  31). 


0    B 


Fig.  126  illustrates  a  condition  in  which  the 
given  plane  iV  is  parallel  with  GrL  :  hence,  the 
auxiliary  line  B,  perpendicular  to  N',  is  parallel 
to  P,  thus  requiring  a  profile  projection  to 
determine  its  traces.  VS  is  the  vertical  trace 
of  the  plane  of  lines  A  and  B  (its  other  trace 
is  not  necessary),  and  angle  e'c'f  is  the  required 
angle  between  line  A  and  .plane  N. 

8i.  To  determine  the  angle  between  a  line 
and  the  coordinate  planes. 

Principle.  This  is  a  special  case  of  Art. 
80,  page  54. 


56 


DESCRIPTIVE   GEOMETRY 


1st  Method.  1.  Revolve  the  line  about  its 
liorizontal  projection  as  an  axis  to  obtain  the 
angle  which  it  makes  with  H.  2.  Revolve 
the  line  about  its  vertical  projection  as  an 
axis  to  obtain  the  angle  which  the  line  makes 
with  V. 

Construction.  Fig.  127  represents  the 
given  line  A  when  revolved  into  H  and 
measures  a,  the  true  angle  which  the  line 
makes  with  H.  Fig.  127  also  represents  the 
line  as  revolved  into  Fand  measuring  /9,  the 
true  angle  which  the  line  makes  with  V 
(Art.  41,  page  28). 

2nd  Method.  1.  Revolve  the  line  parallel 
to  F'to  obtain  the  angle  which  the  line  makes 
with  H.  2.  Revolve  the  line  parallel  to  H  to 
obtain  the  angle  which  the  line  makes  with  V. 

Construction.  Fig.  128  represents  the 
given  line  A  when  revolved  parallel  to  F'and 
measures  a,  the  true  angle  which  the  line 
makes  with  H.  Fig.  128  also  represents  the 
line  A  as  revolved  parallel  to  H  and  measur- 
ing /3,  the  true  angle  which  the  line  makes 
with  V. 


82.  To  determine  the  projections  of  a  line  of 
definite  length  passing  through  a  given  point 
and  making  given  angles  with  the  coordinate 
planes. 

Principle.  This  is  the  converse  of  Art. 
81,  page  55,  and  there  may  be  eight  solutions; 
but  the  sum  of  the  angles,  a  and  /3,  which  the 
line  makes  with  the  coordinate  planes,  cannot 
be  greater  than  90°. 

Construction.  Fig.  128.  Let  it  be  re- 
quired to  draw  a  line  through  point  h  having 
a  length  equal  to  x  and  making  angles  of  a 
and  /3  with  H  and  V^  respectivel3^  Through 
5*"  draw  Jfdl  equal  to  x  and  making  angle  a 
with  CrL\  also  6*4  parallel  to  GL.  These 
projections  will  represent  the  revolved  posi- 
tion of  the  line,  making  its  required  angle 
a  with  H.  Similarly  draw  the  projections 
h^c\,  h'^'c\  to  show  the  revolved  position  and 
I'equired  angle  ^  with  V. 

In  counter-revolution  about  a  vertical  axis 
through  5,  all  possible  horizontal  projections 
of  line  A  will  be  drawn  from  6''  to  a  line 
through  c?2  and  parallel  to  CrL.     But  all  pos- 


ANGLE  BETWEEN  PLANES 


57 


iJ  I 


sible  horizontal  projections  must  be  drawn 
from  fi''  to  the  arc  c\c^\  hence,  c'*,  the  intersec- 
tion of  this  arc  and  the  parallel  through  c^, 
will  be  the  horizontal  projection  of  the  other 
extremity  of  the  required  line.  Since  the 
vertical  projection  of  this  point  must  also  lie 
in  the  parallel  to  CrL  through  Cj,  the  line  is 
definitely  determined. 


J)" 


\ 
\ 
\ 

.""  V' 

f 

\ 

y^ 

1 

\^ 

1 

1 

^83.  To  determine  the  angle  between  two 
planes. 

Principle,  If  a  plane  be  passed  perpendic- 
ular to  the  edge  of  a  diedral  angle,  it  intersects 
the  planes  of  the  diedral  in  lines,  the  angle 
between  which  is  known  as  the  plane  angle  of 
the  diedral.  The  diedral  angle  between  two 
planes  is  measured  by  its  plane  angle. 

Method.  1.  Determine  the  plane  angle 
of  the  given  diedral  angle  by  passing  an 
auxiliary  plane  perpendicuhir  to  the  line  of 
intersection  between  the  two  given  planes, 
and,  therefore,  perpendicular  to  each  of  these 
given  planes.  2.  Find  the  lines  of  intersec- 
tion between  this  auxiliary  plane  and  the 
given  planes.  3.  Determine  the  true  size  of 
this  plane  angle. 

84.  Case  1.  When  it  is  required  to  de- 
termine the  angle  between  any  two  oblique 
planes. 

Construction.  Figs.  129  and  130.  The 
lettering  of  these  two  figures  is  identical 
although  the  diedral  angles  differ. 


58 


DESCRIPTIVE  GEOMETRY 


Pass  the  auxiliary  plane  S  perpendicular  to 
A^  the  line  of  intersection  between  the  planes 
iV  and  L.  Only  one  trace  of  plane  S  is  neces- 
sary for  the  solution  oi.  the  problem  and  in 
this  example  the  horizontal  trace  has  been 
selected.  Then  HiS  must  be  perpendicular 
to  A'^  (Art.  G2,  page  44).  Its  intersection 
with  the  horizontal  traces  of  the  given  planes 
will  determine  points  /  and  e,  one  in  each 
line  of  intersection  between  the  auxiliary  and 
given  planes.  The  point  d,  which  is  com- 
mon to  both  lines  of  intersection,  may  be 
obtained  as  follows:  The  horizontal  project- 
ing plane  of  A  cuts  the  auxiliary  plane  in 
dc,  a  line  perpendicular  to  A.  By  revolving 
A  into  II  the  revolved  position  of  do  may  be 
drawn,  as  at  d'c,  and  by  counter-revolutio* 
point  d  obtained.  Next  revolve  de  and  df 
into  ff  to  measure  the  angle  between  them, 
which  is  the  required  diedral  angle  ed"f. 

One  example  of  this  problem  which  is  com- 
monly met  in  practice  is  here  shown  in  solu- 
tion in  Fig.  131.  Let  iV^  and  L  be  the  given 
planes,  A  their  line  of  intersection,  and  plane 


jS  the  plane  passed  perpendicular  to  A.  Then 
angle  e''d"f^  is  the  true  size  of  the  angle  be- 
tween planes  N  and  L. 

85.  Another  method  for  determining  the 
size  of  the  angle  between  two  planes  is  illus- 
trated by  Fig.  132.  This  is  a  pictorial  repre- 
sentation of  two  intersecting  planes,  iVand  M. 
From  any  point  in  space,  as  point  a,  two  lines 
are  dropped  perpendicular,  one  to  each  plane. 
The  angle  between  these  perpendiculars  is  the 
measurement  of  the  angle  between  planes  iV 
and  i2,  or  its  supplement. 

86.  Case  2.  When  it  is  required  to  deter- 
mine the  angle  between  an  inclined  plane  and 
either  coordinate  plane. 

Construction.  Figs.  133  and  134.  Let 
it  be  required  to  determine  the  true  size  of 
the  angle  between  planes  iVand  H.  Pass  the 
auxiliary  plane  X perpendicular  to  both  iVand 
JI,  and,  therefore,  perpendicular  to  their  line  of 
intersection,  JIJV.  Then  J£X  and  VX  are 
perpendicular  respectively  to  ^iVand  GL. 

The  angle  between  the  lines  ab  and  IIX,  in' 
which  plane  X  intersects  planes  N  and  H  re- 


ANGLE  BETWEEX  PLANES 


59 


spectively,  is  the  required  angle,  the  true  size  mum  inclination  of  plane  N  with  H. 

of  which,  a,  is  determined  by  revolving  the  Suppose  it  is  required  to  determine  the  true 

triangle  containing  it  into  f'^  about  FXasan  size   of   the   angle  between  planes  iV  and   V. 

axis,  or  into  ^  about  ffX  us  an  axis.  Determine  the  line  of  maximum  inclination  of 

The   line  ab  is  known  as  the  line  of  maxi-  plane  N  with  F,  and  its  angle  with  V. 


t 

A 

/* 

f 

0* 

/ 

r 

/ 

\J^ 

s. 

--^"Z^ 

(f^^^ 

• 

/Fig. 

131. 

Fig>133.  ^a 


60 


DESCRIPTIVE  GEOMETRY 


87.  To  determine  the  bevels  for  the  correct 
cuts,  the  lengths  of  hip  and  jack  rafters,  and 
the  bevels  for  the  purlins  for  a  hip  roof. 

Fig.  135  represents '  an  elevation  and  plan 
of  a  common  type  of  hip  roof  having  a  pitch 

equal  to  and  a  width  of  c'J".     ah  is  the 

center  line  of  the  hip  rafter ;  E  vg,  a  cross  sec- 
tion of  the  ridge  ;  F,  K^  X,  M^  are  jack  rafters. 

Hip.  To  find  the  true  length  of  the  hip 
rafter  gJ,  and  the  following  angles: 

Down  cut,  I:  The  intersection  of  the  hip 
with  the  ridge. 

Heel  cut,  2:  The  intersection  of  the  hip 
with  the  plate. 

Side  cut,  3:  Intersection  of  the  hip  with 
the  ridge. 

Top  bevel  of  hip,  4. 

The  true  length  of  the  hip  eh  may  be  ob- 
tained by  revolving  it  parallel  to  F^  as  at  g^6^ 
or  parallel  to  JjT,  as  at  e^^.  The  down  cut 
bevel,  5,  is  obtained  at  the  same  time.  The 
bevel  of  the  top  edge  of  hip  is  found  by  pass- 
ing a  plane  perpendicular   to  ab  intersecting 


the  planes  of  the  side  and  end  roofs.  HZ  is 
the  horizontal  trace  of  this  plane,  and  the 
bevel,  4,  is  obtained  as  in  Art.  84,  page  57. 

Jack  Rafters.  The  down  cut  bevel,  5, 
and  the  heel  cut  bevel,  ^,  of  the  jack  rafters 
are  shown  in  their  true  values  in  the  elevation  ; 
and  the  side  cut,  7,  is  shown  in  the  plan. 
The  true  lengths  of  the  jack,  rafters  are  ob- 
tained by  extending  the  planes  of  their  edges 
to  intersect  the  revolved  position  of  the  hip 
rafter,  as  at  WjO*. 

Purlin.  It  is  required  to  determine  the 
down  cut,  5,  side  cut,  9,  and  angle  between  side 
and  end  face  of  purlin,  10.  To  obtain  the 
down  cut  revolve  side  face  parallel  to  ^and 
the  true  angle,  5,  will  be  obtained.  Similarly, 
the  side  cut  made  on  the  top  or  bottom  face 
is  obtained  by  revolving  that  face  parallel  to 
^,  the  true  angle  being  indicated  by  bevel,  9. 
In  order  to  obtain  the  angle  between  the  side 
and  end  faces,  the  planes  of  which  are  indi- 
cated in  the  figure  by  ^S*  and  ^,  find  the  inter- 
section between  these  planes  and  determine 
the  angle  as  in  Art.  84,  page  57. 


ANGLE  BETWEEN   PLANES 


61 


Fig.  135. 


62 


DESCRIPTIVE  GEOMETRY 


88.  Given  one  trace  of  a  plane,  and  the  angle 
between  the  plane  and  the  coordinate  plane,  to 
determine  the  other  trace. 

There  must  be  two  6ases,  as  follows : 

Case  1.  Given  the  trace  on  one  coordinate 
plane  and  the  angle  which  the  plane  makes 
with  the  same  coordinate  plane. 

Construction.  Fig.  136.  Let  HT  be  the 
given  trace  and  a  the  given  angle  betAveen  T 
and  S.  Draw  A*  perpendicular  to  HT,  it  being 
the  horizontal  projection  of  the  line  of  maxi- 
mum inclination  with  H  (Art.  86,  page  58). 
If  this  line  be  revolved  into  H,  it  will  make 
the  angle  a  with  A'^,  and  d'  will  be  the  resolved 
position  of  the  vertical  trace  of  the  line  of 
maximum  inclination  with  H.  The  vertical 
projection  of  this  trace  must  lie  on  the  vertical 
trace  of  the  horizontal  projecting  plane  of  A 
and  at  a  'distance  from  CiL  equal  to  d'^d'. 
Therefore,  d"  will  be  a  point  in  VT,  the  re- 
quired trace. 

There  may  be  two  solutions,  as  VT  may  be 
above  or  below  GL. 

89.  Case  2.     Given  the  trace  on  one  coor- 


dinate plane  and  the  angle  which  the  plane 
makes  with  the  other  coordinate  plane. 

Construction.  Fig.  137.  Let  ^y  be  the 
given  trace  and  ^  the  given  angle  between  T 
and  V. 

Consider  B,  the  line  of  maximum  inclina- 
tion with  V,  as  revolved  about  the  horizontal 
trace  of  its  vertical  projecting  plane  and  mak- 
ing an  angle,  /8,  with  GiL.  d'  will  be  the 
revolved  position  of  the  vertical  trace  of  B. 
From  e,  with  radius  ed',  describe  arc  d'd'^.  VT, 
the  required  trace,  will  be  tangent  to  this  arc. 
There  may  be  two  solutions,  as  VT  may  be 
above  or  below  CrL. 

90.  To  determine  the  traces  of  a  plane, 
knowing  the  angles  which  the  plane  makes  with 
both  coordinate  planes. 

Construction.  Fig.  138.  Let  it  be  re- 
quired to  construct  the  traces  of  plane  T,  mak- 
ing an  angle  /3  with  iTand  angle  a  with  H. 
The  sum  of  a  and  y8  must  not  be  less  than 
90°  nor  more  than  180°.  Conceive  the  re- 
quired plane  T  as  being  tangent  to  a  sphere, 
the   center   of   which,  c,  lies   in    CrL.     From 


TO  DETERMINE  TRACES  OF  PLANES 


63 


Fig.  138 


the  point  of  tangency  of  the  sphere  and  plane 
T,  conceive  to  be  drawn  the  lines  of  maximum 
inclination  with  ^and  V.  On  revolving  the 
line  of  maximum  inclination  with  V,  into  JJ", 
about  the  horizontal  axis  of  the  sphere,  as  an 
axis,  it  will  continue  tangent  to  the  sphere, 
as  at  A\  making  an  angle  with  GL  equal  to 
the  required  angle  (between  V  and  T'),  and 
its  horizontal  trace  will  lie  in  the  axis  of 
revolution  at  a*,  its  vertical  trace  lying  in  the 
circle  described  from  c  as  a  center,  with  a 
radius  cm'.  Similarly  revolve  the  line  of 
maximum  inclination  with  5^  into  F^  as  at  B\ 
making  an  angle  with  GL  equal  to  the  re- 
quired angle  (between  J^and  T},  and  its  ver- 
tical trace  will  lie  in  the  axis  of  revolution  at 
i*,  its  horizontal  trace  lying  in  the  circle  de- 
scribed from  c  as  a  center,  with  a  radius  ce'. 
Then  ffT  will  contain  a*  and  be  tangent  to 
are  0*^^,  and  VT  will  contain  Jif  and  be  tan- 
gent to  arc  k'tn'.  The  traces  must  intersect 
GL  in  the  same  point.  Both  HT  and  VT 
may  be  either  above  or  below  GL. 

91.    In  Art.  48,  page  34,  reference  was  made 


64 


DESCRIPTIVE  GEOMETRY 


to  a  fourth  construction  for  counter-revolution. 
This  construction  involves  the  angle  of  maxi- 
mum inclination  between  the  oblique  and 
coordinate  planes.  Let  it  be  required  to  draw 
the  projections  of  a  regular  pentagon  lying  in 
plane  JSf,  Fig.  139,  when  its  revolved  position 
is  known.  Pass  plane  X  perpendicular  to 
both  JV  and  F",  intersecting  plane  i\r  in  the 
line  of  maximum  inclination  with  V,  shown  in 
revolved  position  as  gf'.  Since  this  line  shows 
in  its  true  length,  all  distances  on  plane  N 
perpendicular  to  FiY  may  be  laid  off  on  it. 
Then  ge"  represents  the  distance  from  point  e 
to  FTV",  and  e*  will  lie  on  a  line  through  e" 
parallel  to  FTV.  Likewise  other  points  of  the 
pentagon  are  determined.  Also  e"k  is  the 
distance  from  point  e  in  space  to  V;  hence,  e'' 
will  lie  at  a  distance  from  GrL  equal  to  e"k. 


Fig.  139. 


CHAPTER    III 

GENERATION    AND    CLASSIFICATION    OF 
SURFACES 


92.  Every  surface  may  be  regarded  as  hav- 
ing been  generated  by  the  motion  of  a  line, 
which  was  governed  by  some  definite  law. 
The  moving  line  is  called  the  generatrix,  and 
its  dififerent  positions  are  called  elements  of 
the  surface.  Any  two  successive  positions  of 
the  generatrix,  having  no  assignable  distance 
between  them,  are  called  consecutive  elements. 
The  line  which  may  direct  or  govern  the  gen- 
eratrix is  called  the  directrix. 

93.  Surfaces  are  classified  according  to  the 
form  of  the  generatrices,  viz.  : 

Ruled  Surfaces,  or  such  as  may  be  gen- 
erated by  a  rectilinear  generatrix. 


Double-curved  Surfaces,  or  such  as  must 
be  generated  by  a  curvilinear  generatrix. 
These  have  no  rectilinear  elements. 

Tiie  ruled  surfaces  are  reclassified  as  dei^el- 
opable,  and  nondevelopable  or  warped  surfaces. 


\Rvded 


SUBFACES 


Plane 

Single-curved  (developable). 

Cylinder 

Cone 

Convolute 
Warped  (nondevelopable) 
[Double-curved  (nondevelopable) 


94.   Ruled  Surfaces.     A  right  line  may  move 
so  that  all  of  its  positions  will  lie  in  the  same 


6& 


66 


DESCRIPTIVE  GEOMETRY 


plane ;  it  may  move  so  that  any  two  consecu- 
tive elements  will  lie  in  the  same  plane ;  or  it 
may  move  so  that  any  two  consecutive  ele- 
ments will  not  lie  inHhe  same  plane.  Thus, 
ruled  surfaces  are  subdivided  into  three 
classes,  as  follows : 

Plane  Surfaces  :  All  the  rectilinear  ele- 
ments lie  in  the  same  plane. 

Single-curved  Surfaces:  Any  two  con- 
secutive rectilinear  elements  lie  in  the  same 
plane,  i.e.  they  intersect  or  are  parallel. 

Warped  Surfaces:  No  two  consecutive 
rectilinear  elements  lie  in  the  same  plane,  i.e. 
they  are  neither  intersecting  nor  parallel. 

95.  Plane  Surfaces  are  all  alike.  The  rec- 
tilinear generatrix  may  move  so  as  to  touch 
one  rectilinear  directrix,  remaining  always 
parallel  to  its  first  position ;  so  as  to  touch 
two  rectilinear  directrices  which  are  parallel  to 
each  other,  or  which  intersect ;  or  it  may  re- 
volve about  another  right  line  to  which  it  is 
perpendicular. 

96.  Single-curved  Surfaces  may  be  divided 
into  three  classes,  as  follows : 


Cones:  In  which  all  the  rectilinear  ele- 
ments intersect  in  a  point,  called  the  apex. 

Cylinders:  In  which  all  the  rectilinear 
elements  are  parallel  to  each  other.  The 
cylinder  may  be  regarded  as  being  a  cone  with 
its  apex  infinitely  removed. 

CoNVOLUTES  :  In  which  the  successive  rec- 
tilinear elements  intersect  two  and  two,  no 
three  having  one  common  point. 

97.  Conical  Surfaces  are  generated  by  the 
rectilinear  generatrix  moving  so  as  always  to 
pass  through  a  fixed  point,  called  the  apex, 
and  also  to  touch  a  given  curve,  called  the 
directrix.  Since  the  generatrix  is  indefinite  in 
length,  the  surface  is  divided  at  the  apex  into 
two  parts,  called  nappes.  The  portion  of  a 
conical  surface  usually  considered  is  included 
between  the  apex  and  a  plane  which  cuts  all 
the  elements.  This  plane  is  called  the  base  of 
the  cone  and  the  form  of  its  curve  of  inter- 
section with  the  conical  surface  gives  a  dis- 
tinguishing name  to  the  cone,  as,  circular, 
elliptical,  parabolic,  etc.  If  the  base  of  the 
cone    has    a    center,   the   right  line   passing 


SINGLE  CURVED  SURFACES 


67 


through  this  center  and  the  apex  is  called  the 
axis  of  the  cone  (Fig.  I-IO). 

A  Right  Cone  is  one  having  its  base  per- 
pendicular to  its  axis. 


Fig.  140. 


Fig.  141. 


98.  Cylindrical  Surfaces.  The  cylinder  is 
that  limiting  form  of  the  cone  in  which  the 
apex  is  removed  to  infinity.     It  may  be  gen- 


erated by  a  rectilinear  generatrix  which  moves 
so  as  always  to  touch  a  given  curved  directrix, 
having  all  of  its  positions  parallel.  A  plane 
cutting  all  the  elements  of  a  cylindrical  sur- 
face is  called  its  Ja«e,  and  the  form  of  its  curve 
of  intersection  with  the  surface  gives  a  dis- 
tinguishing name  to  the  cylinder,  as  in  the 
case  of  the  cone.  If  the  base  has  a  center, 
the  right  line  through  this  center  parallel  to 
the  elements  is  called  the  axu  (Fig.  141). 

A  cylinder  may  also  be  generated  by  a  cur- 
vilinear generatrix,  all  points  of  which  move  in 
the  same  direction  and  with  the  same  velocity. 

A  Right  Cylinder  is  one  having  its  base 
perpendicular  to  its  axis. 

99.  Convolute  Surfaces  may  be  generated 
by  a  rectilinear  generatrix  which  moves  so  as 
always  to  be  tangent  to  a  line  of  double  cur- 
vature.* Any  two  consecutive  elements,  but 
no  three,  will  lie  in  the  same  plane.  Since 
there  is  an  infinite  number  of  lines  of  double 
curvature,  a  great  variety  of  convolutes  may 

*  A  line  of  double  curvature  is  one  of  which  no  four 
consecutive  points  lie  in  the  same  plane. 


68 


DESCRIPTIVE   GEOMETRY 


exist.  One  such  form  which  may  readily  be 
generated  is  the  helical  convolute  (Fig.  142). 
It  is  the  surface  generated  by  the  hypotenuse 
of  a  right  triangle  under  the  following  condi- 
tions: Suppose  a  right  triangle  of  paper,  or 
some  other  thin,  flexible  material,  to  be  wrapped 
about  a  right  cylinder,  one  leg  of  the  triangle 
coinciding  with  an  element  of  the  cylinder. 
If  the  triangle  be  unwrapped,  its  vertex  will 
describe  the  involute  of  the  base  of  the  cylin- 
der, and  the  locus  of  the  points  of  tangency 
of  its  hypotenuse  and  the  cylinder  will  be  a 
helix,  the  hypotenuse  generating  the  helical 
convolute.  The  convolute  may  also  be  re- 
garded as  being  generated  by  a  rectilinear 
generatrix  moving  always  in  contact  with  the 
involute  and  helix  as  directrices,  and  making 
a  definite  angle  with  the  plane  of  the  involute. 
1 00.  A  Warped  Surface  is  generated  by  a 
rectilinear  generatrix  moving  in  such  a  way 
that  its  consecutive  positions  do  not  lie  in  the 
same  plane.  Evidently  there  may  be  as  many 
warped  surfaces  as  there  are  distinct  laws  re- 
stricting the  motion  of  the  generatrix. 


Any  warped  surface  may  be  generated  by  a 
rectilinear  generatrix  moving  so  as  to  touch 
two  linear  directrices,  and  having  its  consecu- 
tive positions  parallel  either  to  a  given  plane, 
called  a  plane  director,  or  to  the  consecutive 
elements  of  a  conical  surface,  called  a  cone 
director. 

loi.  The  following  types,  illustrated  by 
Figs.  142  to  148,  indicate  the  characteristic 
features  of  warped  surfaces  : 

Hyperbolic  Paraboloid,  Fig.  143.  Two 
rectilinear  directrices  and  a  plane  director,  or 
three  rectilinear  directrices. 

Conoid,  Fig.  144.  One  rectilinear  and  one 
curvilinear  directrix  and  a  plane  director. 

Cylindroid,  Fig.  145.  Two  curvilinear 
directrices  and  a  plane  director. 

Right  Helicoid,  Fig.  146.  Two  curvi- 
linear directrices  and  a  plane  director. 

Oblique  Helicoid,  Fig.  147.  Two  curvi- 
linear directrices  and  a  cone  director. 

Hyperboloid  of  Revolution,  Fig.  148. 
Two  curvilinear  directrices  and  a  cone  direc- 
tor, or  three  rectilinear  directices. 


WARPED  SURFACES 


69 


Fig.  144 


Fig.  147. 


7a 


DESCRIPTIVE   GEOMETRY 


102.  A  Surface  of  Revolution,  Fig.  149,  is 
the  locus  of  any  line,  or  generatrix,  the  posi- 
tion of  which  remains  unaltered  with  reference 
to  a  fixed  right  line  about  which  it  revolves. 
This  fixed  right  line  is  called  the  axis  of  revo- 
lution. A  circle  of  the  surface  generated  by 
any  point  of  the  generatrix  is  called  a  parallel, 
and  planes  perpendicular  to  the  axis  will  cut 
the  surface  in  parallels.  Any  plane  contain- 
ing the  axis  of  revolution  is  called  a  merid- 
ian plane,  and  the  line  cut  from  the  surface 
by  this  plane  is  called  a  meridian  line.  All 
meridian  lines  of  the  same  surface  are  obvi- 
ously identical,  and  any  one  of  them  may  be 
considered  as  a  generatrix.  That  meridian 
plane  which  is  parallel  to  a  coordinate  plane 
is  called  the  principal  meridian. 

If  the  generatrix  be  a  right  line  lying  in 
the  same  plane  as  the  axis,  it  will  either  be 
parallel  with  it  or  intersect  it  ;  in  the  former 
case  the  surface  generated  will  be  a  cylinder, 
in  the  latter,  a  cone,  and  these  are  the  only 
single-curved  surfaces  of  revolution. 

If  the  generatrix  does  not  lie  in  the  same 


plane  with  the  axis,  the  consecutive  positions 
are  neither  parallel  nor  intersecting.  The 
surface  must  then  be  warped  and  its  meridian 
line  will  be  an  hyperbola.  This  is  the  only 
warped  surface  of  revolution.  It  may  also  be 
generated  by  revolving  an  hyperbola  about  its 
conjugate  axis,  and  is  known  as  the  hyperbo- 
loid  of  revolution  of  one  nappe  (Fig.  148). 

103.  Double-curved  Surfaces.  With  the  ex- 
ception of  the  cylinder,  cone,  and  hyperboloid 
of  revolution  all  surfaces  of  revolution  are  of 
double  curvature.  They  are  infinite  in  num- 
ber and  variety.     Representative  types  are  : 

The  Sphere:  Generated  by  revolving  a 
circle  about  its  diameter. 

The  Prolate  Spheroid:  Generated  by 
revolving  an  ellipse  about  its  major  axis. 

The  Oblate  Spheroid  :  Generated  by 
revolving  an  ellipse  about  its  minor  axis. 

The  Paraboloid  :  Generated  by  revolv- 
ing a  parabola  about  its  axis  (Fig.  149). 

The  Hyperboloid  of  Two  Nappes: 
Generated  by  revolving  an  hyperbola  about 
its  transverse  axis. 


DOUBLE-CURVED  SURFACES 


71 


The  Torus  (annular  or  not):  Generated 
by  revolving  a  circle  about  a  line  of  its  plane 
other  than  its  diameter.  Fig.  150  illustrates 
an  annular  torus. 


The  Double-curved  Surface  of  Traxs- 
POsiTiox  —  Serpentine  :  Generated  by  a 
sphere  the  center  of  which  moves  along  an 
helix  (Fig.  151). 


PARABOLOID 
Fig.  149. 


TORUS  (AMMULAfi'} 
Fig.  150. 


SERPENTINE 
Fig.    151, 


*^BR-A^ 


OF 


UNIVERSITY 


CHAPTER   IV 


TANGENT  PLANES 


104.  A  plane  is  tangent  to  a  single-curved 
surface  when  it  contains  one,  and  only  one 
element  of  that  surface.  The  two  lines  com- 
monly determining  a  tangent  plane  are  the  tan- 
gent element  and  a  tangent  to  the  surface  at 
some  point  in  this  element.  If  this  second 
line  lies  in  one  of  the  coordinate  planes  it  will 
be  a  trace  of  the  tangent  plane. 

In  Fig.  152  point  d  is  on  the  surface  of  a 
cone  to  which  a  tangent  plane  is  to  be  drawn. 
The  line  drawn  through  point  d  and  the  apex 
of  the  cone  will  be  the  element  at  which  the 


plane  is  to  be  tangent  and,  therefore,  one  line 
of  the  tangent  plane.  If  a  second  line,  ck,  be 
drawn  through  point  d,  and  tangent  to  any 
section  of  the  cone  containing  this  point,  it 
will  be  a  second  line  of  the  tangent  plane. 
The  traces  of  these  lines  will  determine  JjTaS' 
and  VS,  the  traces  of  the  required  tangent 
plane.  If  the  base  of  the  single-curved  sur- 
face coincides  with  one  of  the  coordinate 
planes,  as  in  Fig.  152,  bk  can  be  used  for 
the  tangent  line,  thus  determining  the  hori- 
zontal trace  directly. 


72 


TANGENT  PLANES 


73 


Fig.  152. 


105.  One  projection  of  a  point  on  a  single- 
curved  surface  being  given,  it  is  required  to 
pass  a  plane  tangent  to  the  surface  at  the  ele- 
ment containing  the  given  point. 

Principle.  The  tangent  plane  will  be  de- 
termined by  two  intersecting  lines,  one  of 
which  is  the  element  of  the  surface  on  which 
the  given  point  lies,  and  the  second  is  a  line 
intersecting  this  element  and  tangent  to  the 
single-curved  surface,  preferably  in  the  plane 
of  the  base. 

Method.  1.  Draw  the  projections  of  the 
element  containing  the  given  point.  2.  In 
the  plane  of  the  base  draw  a  second  line 
tangent  to  the  base  at  the  tangent  element. 
3.    Determine  the  plane  of  these  lines. 

If  the  plane  of  the  base  coincides  with  one 
of  the  coordinate  planes,  the  tangent  line  will 
be  one  of  the  traces  of  the  required  tangent 
plane. 

Note.  In  this  and  the  following  problems 
the  base  of  the  single-curved  surface  is  consid- 
ered as  lying  on  one  of  the  coordinate  planes. 


74 


DESCRIPTIVE  GEOMETRY 


Construction.  Fig.  153.  Let  the  single- 
curved  surface  be  a  cone  which  is  defined  by 
its  projections,  and  tlje  base  of  which  lies  in 
one  of  the  coordinate  planes;  in  this  case  in  H. 
Let  <?•  be  the  vertical  projection  of  the  given 
point.  Through  c"  draw  E"^  the  vertical  pro- 
jection of  the  tangent  element.  The  vertical 
projection  of  the  horizontal  trace  of  this  ele- 
ment is  k"^  which,  being  a  point  in  the  base, 
will  be  horizontally  projected  at  k\  and  k%,  thus 
making  two  possible  tangent  elements,  E^  and 
E^ ;  also  two  possible  locations  for  point  c?, 
and,  therefore,  two  possible  tangent  planes. 
Since  the  base  of  the  cone  lies  in  H,  the 
horizontal  traces,  ^>S'  and  HN  of  the  tangent 
planes  S  and  iV,  will  be  tangent  to  the  base 
at  1(f^  and  ArJ.  The  vertical  traces  of  the 
tangent  elements,  l"  and  o",  determine  the 
necessary  points  in  VS  and  FiV,  the  required 
vertical  traces  of  the  tangent  planes. 

1 06.  To  pass  a  plane  tangent  to  a  cone  and 
through  a  given  point  outside  its  surface. 

Pkinciple.  Since  all  tangent  planes  con- 
tain the  apex  of  the  cone,  the  required  plane 


Fig.   153. 


PLANE  TANGENT  TO  GONE 


75 


must  contain  a  line  drawn  through  the  apex 
of  the  cone  and  the  given  point.  It  must  also 
have  one  of  its  traces  tangent  to  the  base  of 
the  cone,  since  this  base  is  supposed  to  lie  in 
a  coordinate  plane. 

Method.  1.  Draw  the  projections  of  a 
line  passing  througli  the  apex  of  the  cone  and 
the  given  point.  2.  Determine  its  traces. 
3.  Through  that  trace  of  the  auxiliary  line 
which  lies  in  the  plane  of  the  base  draw 
the  trace  of  the  required  plane  tangent  to  the 
base  of  the  cone.  4.  Draw  the  other  trace  of 
the  plane  through  the  other  trace  of  the 
auxiliary  line.  There  are  two  possible  tan- 
gent planes. 

107.  To  pass  a  plane  tangent  to  a  cone  and 
parallel  to  a  given  line. 

Pkixceple.  The  tangent  plane  must  con- 
tain the  apex  of  the  cone  and  a  line  through 
the  apex  parallel  to  the  given  line. 

Method.  1.  Through  the  apex  of  the 
cone  draw  a  line  parallel  to  the  given  line. 
2.    Obtain  the  traces  of   this   auxiliary  line. 


3.  Draw  the  traces  of  the  required  tangent 
plane  through  the  traces  of  the  auxiliary  line, 
making  one  of  them  tangent  to  the  base  of  the 
cone.     There  are  two  possible  tangent  planes. 

108.  To  pass  a  plane  tangent  to  a  cylinder 
and  through  a  given  point  outside  its  surface. 

Princii'LE.  The  tangent  plane  will  be  de- 
termined by  two  intersecting  lines,  one  of 
which  is  drawn  through  the  given  point 
parallel  to  the  elements  of  the  cylinder,  and 
the  second  is  drawn  tangent  to  the  cylinder 
from  some  point  in  the  first  line,  preferably  in 
the  plane  of  the  base. 

Method.  1.  Through  the  given  point 
draw  the  projections  of  a  line  parallel  to  the 
elements  of  the  cylinder.  2.  Determine  the 
traces  of  this  auxiliary  line.  3.  Through 
that  trace  of  the  auxiliary  line  which  lies  in 
the  plane  of  the  base  of  the  cylinder  draw  one 
trace  of  the  required  plane  tangent  to  the  base. 

4.  Through  the  other  trace  of  the  auxiliary 
line  draw  the  second  trace  of  the  tangent 
plane.     There  are  two  possible  tangent  planes. 


76 


DESCRIPTIVE  GEOMETRY 


Construction.  Fig.  154.  Let  a"  and  a^ 
be  the  projections  of  a  given  point  through 
which  the  plane  is  to  be  passed  tangent  to  the 
cylinder.  The  base  of  the  cylinder  rests  on  H. 
B"  and  B''  are  the  projections  of  the  auxiliary 
line  drawn  through  point  a  parallel  to  the  ele- 
ments of  the  cylinder.  Through  d}^  the  hori- 
zontal trace  of  this  line,  HS  and  HN^  the 
horizontal  traces  of  the  two  possible  tangent 
planes,  may  be  drawn  tangent  to  the  base  of 
the  cylinder.  The  vertical  traces  of  these 
planes,  VS  and  VN^  must  contain  c",  the  verti- 
cal trace  of  the  line  B.  The  elements  at  which 
planes  iVand  8  are  tangent  are  lines  F  and  K^ 
respectively. 

109.  To  pass  a  plane  tangent  to  a  cylinder 
and  parallel  to  a  given  line. 

Principle.  The  tangent  plane  must  be 
parallel  to  a  plane  determined  by  the  given 
line  and  a  line  intersecting  it,  which  is  parallel 
to  the  elements  of  the  cylinder. 

Method.  1.  Through  any  point  of  the 
given  line  draw  a  line  parallel  to  the  elements 


PLANE  TANGENT  TO  CYLINDER 


77 


of  the  cylinder.  2.  Determine  the  traces  of 
the  plane  of  these  two  lines.  3.  Draw  the 
traces  of  the  tangent  plane  parallel  to  the 
traces  of  the  auxiliary  plane  (Art.  18,  page 
12),  one  of  the  traces  being  tangent  to  the 
base  of  the  cylinder.  There  may  be  two  tan- 
gent planes. 

CoNSTKUCTiox.  Fig.  155.  Through  any 
point,  c,  of  the  given  line  A,  draw  line  B  paral- 
lel to  the  elements  of  the  cylinder.  The  tan- 
gent planes  will  be  parallel  to  X,  the  plane  of 
these  lines.  N  and  *S'  will  be  the  required 
tangent  planes. 

no.  A  plane  is  tangent  to  a  double-curved 
surface  when  it  contains  one,  and  only  one 
point  of  that  surface.  The  two  lines  com- 
monly determining  the  tangent  plane  are  the 
lines  tangent  to  the  meridian  and  parallel  at 
the  point  of  tangency  (Art.  102,  page  70). 

A  Normal  is  the  line  perpendicular  to  the 
tangent  plane  at  the  point  of  tangency. 

A  Normal  Plane  is  any  plane  containing  the 
normal. 


78 


DESCRIPTIVE  GEOMETRY 


III.  One  projection  of  a  point  on  the  surface 
of  a  double -curved  surface  of  revolution  being 
given,  it  is  required  to  pass  a  plane  tangent  to 
the  surface  at  that  point. 

Principle.  Planes  tangent  to  double- 
curved  surfaces  of  revolution  must  contain 
the  tangents  to  the  meridian  and  parallel  at 
the  point  of  tangency. 

There  may  be  two  methods. 

1st  Method.  1.  Through  the  given  point 
draw  a  meridian  and  a  parallel  (Art.  102, 
page  70).  2.  Draw  tangents  to  these  curves 
at  the  given  point.  3.  Determine  the  plane 
of  these  tangents. 

Construction.  Fig.  156.  Given  the  ellip- 
soid with  its  axis  perpendicular  to  H.  A  plane 
is  required  to  be  drawn  tangent  to  the  surface 
at  the  point  having  e*  for  its  horizontal  pro- 
jection. With  f'e'^  as  a  radius,  and  /''  as  a 
center,  describe  the  circle  which  is  the  hori- 
zontal projection  of  the  parallel  through  e. 
One  of  the  possible  vertical  projections  of  the 
parallel  is  tliat  portion  of  B"  lying  within  tlie 
ellipse.     Project  e^  on  to  this  line  to  obtain  e". 


the  vertical  projection  of  the  given  point  e. 
Through  point  e  draw  line  B  tangent  to  the 
parallel,  and,  therefore,  in  the  plane  of  the 
parallel.  That  portion  of  A''  lying  within 
the  circle,  which  is  the  horizontal  projection 
of  the  ellipsoid,  will  be  the  horizontal  projec- 
tion of  the  meridian  drawn  through  point  e. 
Revolve  this  meridian  line  about  fk  as  an  axis 
until  it  coincides  with  the  principal  meridian 
(Art.  102,  page  70),  the  vertical  projection  of 
which  will  be  shown  by  the  ellipse.  The  re- 
volved position  of  the  vertical  projection  of 
point  e  will  now  be  at  ej,  and  a  line  may  be 
drawn  tangent  to  the  meridian  at  this  point, 
its  projections  being  A\^  A'[.  Counter-revolve 
this  meridian  plane  to  determine  the  true  posi- 
tion of  the  tangent  line,  shown  by  its  projec- 
tions at  A'\  A''.  The  traces  of  the  tangent 
lines  A  and  B  will  determine  the  traces  of  the 
required  tangent  plane.  There  are  two  possi- 
ble tangent  planes. 

2nd  Method.  1.  Draw  the  projections  of 
a  cone  tangent  to  the  double-curved  surface 
of  revolution  at  the  parallel  passing  through 


PLANE  TANGENT  TO  ELLIPSOID 


79 


Fig.  156. 


the  given  point.  2.  Pass  a  plane  tangent 
to  the  auxiliary  cone  at  the  element  drawn 
through  the  given  point  (Art.  105,  page  73). 

Construction.  Fig.  156.  J.*  and  A*"  are 
the  projections  of  the  element  of  a  cone  tan- 
gent to  the  ellipsoid  and  containing  point  e. 
A  plane  tangent  to  the  cone  at  line  A  will  be 
the  required  plane  tangent  to  the  ellipsoid 
at  point  e.  If  the  vertical  trace  of  line  A 
lies  beyond  the  limits  of  the  paper,  the  direc- 
tion of  FiV  may  be  determined  by  observing 
that  the  trace  of  the  required  plane  must  be 
perpendicular  to  the  normal  D  at  the  point  e 
(Art.  110,  page  77).  There  are  two  possible 
tangent  planes. 

112.  Through  a  point  in  space  to  pass  a 
plane  tangent  to  a  double-curved  surface  of 
revolution  at  a  given  parallel. 

^Iethod.  1.  Draw  a  cone  tangent  to  the 
double-curved  surface,  of  revolution  at  the 
given  parallel.  2.  Pass  a  plane  through 
the  given  point  tangent  to  the  cone  (Art. 
106,  page  74).  There  are  two  possible  tangent 
planes. 


80 


DESCRIPTIVE  GEOMETRY 


113.  To  pass  a  plane  tangent  to  a  sphere  at 
a  given  point  on  its  surface. 

Method.  This  may  be  solved  as  in  Art. 
Ill,  or  by  the  following  method.  1.  Through 
the  given  point  draw  a  radius  of  the  sphere. 
2.  Pass  a  plane  through  the  given  point  per- 
pendicular to  the  radius  (Art.  72,  page  50), 
and  this  will  be  the  required  plane. 

114.  Through  a  given  line  to  pass  planes 
tangent  to  a  sphere.* 

Principle.  Conceive  a  plane  as  passed 
through  the  center  of  the  sphere  and  perpen- 
dicular to  the  given  line.  It  will  cut  a  great 
circle  from  the  sphere  and  lines  from  the  re- 
quired tangent  planes.  These  lines  will  be 
tangent  to  the  great  circle  of  the  sphere  and 
intersect  the  given  line  at  the  point  in  which 
it  intersects  the  auxiliary  plane.  The  planes 
determined  by  these  tangent  lines  and  the 
given  line  will  be  the  required  tangent  planes. 

*  For  a  general  solution  of  problems  requiring  the  draw- 
ing of  tangent  planes  to  double-curved  surfaces  of  revolu- 
tion, and  through  a  given  line,  see  Art.  160,  page  122. 


Method.  1.  Through  the  center  of  the 
sphere  pass  a  plane  perpendicular  to  the  given 
line  (Art.  72,  page  50)  and  determine  its 
traces.  2.  Determine  the  point  in  which  this 
plane  is  pierced  by  the  given  line  (Art.  57, 
page  42).  3.  Into  one  of  the  coordinate 
planes  revolve  the  auxiliary  plane  containing 
the  center  of  the  sphere,  the  great  circle  cut 
from  the  sphere,  and  the  point  of  intersection 
with  the  given  line.  4.  From  the  latter 
point  draw  lines  tangent  to  the  revolved 
position*  of  the  great  circle  of  the  sphere. 
These  lines  will  be  lines  of  the  required  tan- 
gent planes.  5.  Counter-revolve  the  auxil- 
iary plane  containing  the  lines  of  the  tangent 
planes.  6.  Determine  the  planes  defined  by 
the  given  line  and  each  of  the  tangent  lines 
obtained  by  4.  These  will  be  the  required 
tangent  planes. 

Construction.  Fig.  157.  The  given  line 
is  A,  and  the  center  of  the  sphere  is  e.  HX 
and  VX  are  the  traces  of  the  auxiliary  plane 
perpendiculjlr   to    A   and   passing  through   e 


PLANE  TANGENT  TO  SPHERE 


81 


(Art.  72,  page  50).  The  point/  is  that  of  the 
intersection  of  line  A  and  plane  X.  Revolve 
plane  X  into  the  vertical  coordinate  plane  and 
e'  will  be  the  revolved  position  of  the  center 
of  the  sphere,  and  f  the  revolved  position  of 
the  point  /.      Draw  the  great  circle  of   the 


sphere  and  the  tangents  C  and  B'. 

In  counter-revolution  these  lines  will  be  at 
C  and  B,  intersecting  J.  at  /.  C  and  A  will 
be  two  intersecting  lines  of  one  of  the  required 
tangent  planes,  S,  and  the  second  plane,  JV,  will 
be  determined  by  lines  A  and  B. 


Fig.  157. 


CHAPTER  V 


INTERSECTION  OF  PLANES  WITH  SURFACES, 
AND  THE   DEVELOPMENT  OF  SURFACES 


115.  To  determine  the  intersection  of  any 
surface  with  any  secant  plane. 

General  Method.  1.  Pass  a  series  of 
auxiliary  cutting  planes  which  will  cut  lines, 
straight  or  curved,  from  the  surface,  and  right 
lines  from  the  secant  plane.  2.  The  inter- 
sections of  these  lines  are  points  in  the  re- 
quired curve  of  intersection. 

This  method  is  applicable  alike  to  prisms, 
pyramids,  cylinders,  cones,  or  double-curved 


surfaces  of  revolution.  The  auxiliary  cutting 
planes  may  be  used  in  •dny  position,  but  for 
convenience  they  should  be  chosen  so  as  to 
cut  the  simplest  curves  from  the  surface,  that 
is,  straight  lines  or  circles. 

With  solids  such  as  prisms,  pyramids, 
single-curved,  or  other  ruled  surfaces,  the 
above  method  consists  in  finding  the  intersec- 
tion of  each  element  with  the  oblique  plane  by 
Art.  61,  page  44. 


82 


INTERSECTION  OF  PLANE  WITH  PYRAMID 


83 


ii6.  A  tangent  to  the  curve  of  intersection 
of  a  jDlane  with  a  single-curved  surface  may  be 
drawn  by  passing  a  plane  tangent  to  the  sur- 
face at  the  point  assumed  (Art.  Ill,  page  78). 
The  line  of  intersection  of  the  tangent  plane 
with  the  secant  plane  will  be  the  required 
tangent. 

117.  The  true  size  of  the  cut  section  may  al- 
ways be  found  by  revolving  it  into  one  of  the 
coordinate  planes,  about  a  trace  of  the  secant 
plane  as  an  axis. 

118.  A  right  section  is  the  section  cut  from 
the  surface  by  a  plane  perpendicular  to  the 
axis. 

119.  The  development  of  a  surface  is  its 
true  size  and  shape  when  spread  open  upon  a 
plane.  Only  surfaces  having  two  consecutive 
elements  in  the  same  plane  can  be  developed, 
as  only  such  surfaces  can  be  made  to  coincide 
with  a  plane.  Therefore,  only  single-curved 
surfaces,  and  solids  bounded  by  planes,  can  be 
developed.  Solids  bounded  by  planes  are 
developed  by  finding  the  true  size  and  shape 


of  each  successive  face.  Single-curved  sur- 
faces are  developed  by  placing  one  element  in 
contact  with  the  plane  and  rolling  the  surface 
until  every  element  has  touched  the  plane. 
That  portion  of  the  plane  covered  by  the  sur- 
face in  its  revolution  is  the  development  of 
the  surface, 

120.  To  determine  the  intersection  of  a  plane 
with  a  pyramid. 

Principle.  Since  a  pyramid  is  a  solid 
bounded  by  planes,  the  problem  resolves  itself 
into  determining  the  line  of  intersection  be- 
tween two  planes.  Again,  since  the  pyramid 
is  represented  by  its  edges,  the  problem  still 
further  resolves  itself  into  determining  the 
points  of  piercing  of  these  edges  with  the 
plane. 

Method.  1.  Determine  the  points  in  which 
the  edges  of  the  given  pyramid  pierce  the 
given  plane.  2.  Connect  the  points  thus 
obtained  in  their  order,  thereby  determining 
the  required  intersection  between  the  plane 
and  pyramid. 


84 


DESCRIPTIVE  GEOMETRY 


Construction.  Fig.  158.  Given  the  pyra- 
mid of  which  lines  A,  B^ .  (7,  D,  and  E  are  the 
lateral  edges,  or  elemenjts,  intersected  by  plane 
iV^.  The  points  of  piercing,  a,  6,  e,  d,  and  e, 
of  the  elements  with  plane  iV  have  been  de- 
termined by  the  use  of  the  horizontal  project- 
ing planes  of  the  elements  (Art.  59,  page  42), 
and  the  lines  ab,  be,  cd,  de,  and  ea,  joining 
these  points  of  piercing  in  their  order,  are  the 
lines  of  intersection  of  plane  and  pyramid. 

Since  all  the  auxiliary  planes  used  contain 
point  I,  the  apex  of  the  pyramid,  and  are  per- 
pendicular to  ff,  they  must  contain  a  line 
through  I  perpendicular  to  JI;  hence,  o,  the 
point  of  piercing  of  this  line  and  plane  JV,  is 
a  point  common  to  all  the  lines  of  intersection 
between  plane  iV  and  the  auxiliary  planes. 
By  observing  this  fact  the  work  of  construc- 
tion can  be  slightly  shortened. 

The  true  size  of  the  cut  section  is  obtained 
by  revolving  each  of  its  points  into  V  about 
FiVas  an  axis  (Art  43,  page  31). 

121.    To  develop  the  pyramid. 

Principle.  If  the  pyramid  be  laid  on  a 
plane  and  be  made  to  turn  on  its  edges  until 


each  of  its  faces  in  succession  has  come  into 
contact  with  the  plane,  that  portion  of  the 
plane  which  has  been  covered  by  the  pyramid 
in  its  revolution  will  be  the  development  of 
the  pyramid.  From  the  above  it  is  evident 
that  every  line  and  surface  of  the  pyramid 
will  appear  in  its  true  size  in  development. 

Method.  1.  Determine  the  true  length 
of  each  line  of  the  pyramid.  2.  Construct 
each  face  in  its  true  size  and  in  contact  with 
adjacent  faces  of  the  pyramid. 

Construction.  Figs.  158  and  159.  The 
true  lengths  of  the  elements  and  their  seg- 
ments have  been  determined  by  revolving 
them  parallel  to  F",  as  at  ^j,  B^,  etc.,  a^,  5j,  etc., 
Fig.  158.  The  edges  F,  Gr,  J,  J",  and  K  are 
already  shown  in  their  true  lengths  in  their 
horizontal  projections,  since  the  plane  of  these 
lines  is  parallel  to  H.  Hence,  through  any 
point  I,  Fig.  159,  representing  the  apex,  draw 
a  line  -B,  equal  in  length  to  B^.  With  Z  as  a 
center  and  with  a  radius  equal  to  Cj,  draw  an 
arc  of  indefinite  length.  With  the  end  of  B 
as  a  center  and  with  a  radius  equal  to  (T^ 
draw  an  arc  intersecting  the  first  in  point  w, 


DEVELOPMENT  OF  PYRAMID 


85 


thus  definitely  locating  lines  (7  and  Cr.     The 

other  faces  are  determined  in  like  manner. 

The  development  of  the  cut  is  obtained  by 

^^  laying   off   from  Z,  on  its  corresponding   ele- 

^\        ment,  the  true  lengths  of  the  elements  from 

\^  \^    the  apex  of  the  pyramid  to  the  cut  section, 

and  joining  the  points  thus  obtained. 

The  base  and  cut  section  may  be  added  to 
complete  the  development  of  the  pyramid. 


Fig.  159 


86 


DESCRIPTIVE   GEOMETRY 


122.  To  determine  the  curve  of  intersection 
between  a  plane  and  any  cone. 

Principle.  The  problem  is  identical  with 
that  of  the  intersection  of  a  plane  with  a  pyra- 
mid, for  a  cone  may  be  considered  as  being  a 
pyramid  of  an  infinite  number  of  faces. 

Method.  1.  Pass  a  series  of  auxiliary 
planes  perpendicular  to  one  of  the  coordinate 
planes  and  cutting  elements  from  the  cone. 
2.  Each  auxiliary  plane,  save  the  tangent 
planes,  will  cut  two  elements  from  the  cone 
and  a  right  line  from  the  given  plane.  The 
intersections  of  this  line  with  the  elements 
give  two  points  in  the  required  curve. 

Construction.  Fig.  160.  Let  it  be  re- 
quired to  determine  the  curve  of  intersection 
between  plane  N  and  the  oblique  cone  with 
its  circular  base  parallel  to  H.  Pass  a  series 
of  auxiliary  cutting  planes  through  the  cone, 
containing  its  apex,  a,  and  perpendicular  to  H. 
Plane  X  is  one  sach  plane  which  cuts  two  ele- 
ments, ah  and  ac,  from  the  cone,  and  the  line 
B  from  plane  N.  B  intersects  element  ac  in 
point  e  and  element  ah  in  point  /,  and  these 
are  two  points  of  the  required  curve  of  inter- 


section between  plane  iVand  the  cone.  Simi- 
larly determine  a  sufficient  number  of  points 
to  trace  a  smooth  curve.  The  planes  passing 
through  the  contour  elements  of  each  pro- 
jection should  be  among  those  chosen. 

As  in  the  case  of  the  pyramid,  the  work  of 
construction  may  be  slightly  shortened  by 
observing  that  since  all  of  the  auxiliary  planes 
are  perpendicular  to  H^  and  pass  through  the 
apex  of  the  cone,  they  all  contain  the  line 
passing  through  the  apex  and  perpendicular 
to  H.  This  line  pierces  plane  iV^in  (f,  a  point 
common  to  all  lines  of  intersection  between  N 
and  the  auxiliary  planes. 

The  true  size  of  the  cut  section  is  deter- 
mined by  revolving  each  of  its  points  into  V 
about  FiVas  an  axis. 

123.  To  determine  the  development  of  any- 
oblique  cone. 

Principle.  When  a  conical  surface  is 
rolled  upon  a  plane,  its  apex  will  remain 
stationary,  and  the  elements  will  successively 
roll  into  contact  with  the  plane,  on  which  they 
will  be  seen  in  their  true  lengths  and  at  their 
true  distances  from  each  other. 


DEVELOPMENT  OF  CX)NE 


87 


Construction.  Since  in  Fig.  160  the  base 
of  the  cone  is  parallel  to  J?",  the  true  distances 
between  the  elements  may  be  measured  upon 
the  circumference  of  the  base ;  therefore,  to 
develop  the  conical  surface  upon  a  plane, 
through  any  point  a,  Fig.  161,  representing 
the  apex,  draw  a  line  ac,  equal  in  length  to 
a'^Cj,  the  revolved  position  of  element  oc.  Fig. 
160.  With  a  as  a  center,  and  with  a  radius 
equal  to  the  true  length  of  the  next  element, 
ak^  draw  an  arc  of  indefinite  length.  With  c 
as  a  center,  and  with  a  radius  equal  to  (^V"^ 
draw  an  arc  intersecting  the  first  in^.  This 
process  must  be  repeated  until  the  complete 


DESCRIPTIVE   GEOMETRY 


development  has  been  found.  The  accuracy 
of  the  development  depends  upon  the  number 
of  elements  used,  the  g;reater  number  giving 
greater  accuracy. 

The  development  of  the  curve  of  intersection 
is  obtained,  as  in  the  pyramid,  by  laying  off 
from  the  apex,  on  their  corresponding  ele- 
ments, the  true  lengths  of  the  portions  of  the 
elements  from  the  apex  to  the  cut  section, 
and  joining  the  points  thus  found. 

124.  To  determine  the  curve  of  intersection 
between  a  plane  and  any  cylinder. 

Principle.  A  series  of  auxiliary  cutting 
planes  parallel  to  the  axis  of  the  cylinder  and 
perpendicular  to  one  of  the  coordinate  planes 
will  cut  elements  from  the  cylinder  and  right 
lines  from  the  given  plane.  The  intersections 
of  these  elements  arid  lines  will  determine 
points  in  the  required  curve. 

Construction.  Fig.  162.  Given  the  ob- 
lique elliptical  cylinder  cut  by  the  plane  iV. 
Plane  X  is  one  of  a  series  of  auxiliary  cutting 
planes  parallel  to  the  axis  of  the  cylinder  and 
perpendicular  to  H.     Since  it   is   tangent  to 


the  cylinder,  it  contains  but  one  element,  A^ 
and  intersects  the  given  plane  in  the  line  (r. 
Since  lines  G-  and  A  lie  in  plane  X,  they  in- 
tersect in  a,  one  point  in  the  required  curve 
of  intersection  between  the  cylinder  and  the 
given  plane  N.  Likewise  plane  Z  intersects 
the  cylinder  in  elements  C  and  D,  and  the 
plane  iV^in  line  K,  the  intersections  of  which 
with  lines  C  and  D  are  c  and  c?,  two  other 
points  in  the  required  curve. 

The  true  size  of  the  cut  section  has  been 
determined  by  revolving  it  into  V  about  VN 
as  an  axis. 

125.    To  develop  the  cylinder. 

Principle.  When  a  cylinder  is  rolled  upon 
a  plane  to  determine  its  development,  all  the 
elements  will  be  shown  parallel,  in  their  true 
lengths,  and  at  their  true  distances  from  each 
other.  Since  in  an  oblique  cylinder  the  bases 
will  unroll  in  curved  lines,  it  is  necessary  to 
determine  a  right  section  which  will  develop 
into  a  right  line,  and  upon  which  the  true 
distances  between  the  elements  may  be  laid  off. 
This  line  will  be  equal  to  the  periphery  of  the 


DEVELOPMENT  OF  CYLINDER 


89 


right  section,  and  the  elements  will  be  per- 
pendicular to  it.  The  ends  of  these  perpen- 
diculars will  be  at  a  distance  from  the  line 
equal  to  the  true  distances  of  the  ends  of  the 
elements  from  the  right  section.  A  smooth 
curve  may  then  be  drawn  through  the  ends  of 
the  perpendiculai-s. 

Method.  1.  Draw  a  right  line  equal  in 
length  to  the  periphery  of  the  right  section. 
2.  Upon  this  right  line  lay  off  the  true  dis- 
tances between  the  elements.  3.  Through 
the  points  thus  obtained  draw  perpendiculars 
to  the  right  line.  4.  On  these  perpendiculars 
lay  off  the  true  lengths  of  the  corresponding 
elements,  both  above  and  below  the  right  line. 
5.  Trace  a  smooth  curve  through  the  ends  of 
the  perpendiculars. 

Construction.  Figs.  162  and  163.  The 
secant  plane  Noi  Fig.  162  has  been  so  chosen 
as  to  cut  a  right  section  from  the  cylinder, 
that  is,  the  traces  of  the  plane  are  perpen- 
dicular to  the  projections  of  the  axis  of  the 
cylinder  (Art.  62,  page  44).  Element  D  has 
been  revolved  parallel  to  V  to  obtain  its  true 


90 


DESCRIPTIVE  GEOMETRY 


Fig.  163. 


DEVELOPMENT  OF  CYLINDER 


91 


length  i)j,  and  d"  is  projected  to  d^,  thus  oh- 
taining  the  true  lengths  of  each  portion  of  D. 
Since  all  the  elements  of  a  cylinder  are  of 
the  same  length,  2)j  represents  the  true  length 
of  each  element,  and  their  segments  are  ob- 
tained by  projecting  the  various  points  of  the 
cut  section  upon  it,  as  at  Jj,  <?j,  etc.  Upon  the 
right  line  dd.  Fig.  163,  the  true  distances 
between  the  elements,  dc,  cb,  etc.,  have  been 
laid  off  equal  to  the  rectified  distances  <iV, 
c'b',  etc.  Through  points  <£,  c,  5,  etc.,  perpen- 
diculars to  line  dd  have  been  drawn  equal  to 
the  true  length  of  the  elements.  The  portions 
above  and  below  dd  are  equal  to  the  lengths 
of  the  elements  above  and  below  the  cut 
section. 

126.  If  the  axis  of  the  cylinder  be  parallel 
to  a  coordinate  plane,  tlie  development  may  be 
obtained  without  the  use  of  a  right  section. 

Fig.  164  represents  an  oblique  cylinder 
with  its  axis  parallel  to  V  and  its  bases 
parallel  to  Jff.  The  elements  are  represented 
in  vertical  projection  in  their  true  lengths, 
and  in  horizontal  projection  at  their  true 
distances  apart    measured    on    the  periphery 


of,  the  base.  If  the  cylinder  be  rolled  upon 
a  plane,  the  ends  of  the  elements  will  move  in 
planes  perpendicular  to  the  elements ;  there- 
fore, b'  will  lie  on  b^b'  perpendicular  to  5',  and 
at  a  distance  from  a'  equal  to  a*J*.  Likewise 
(/  will  lie  on  cV  perpendicular  to  C*  and  at  a 
distance  from  b'  equal  to  6*c*,  etc.  The  figure 
shows  but  one  half  the  development. 


92 


DESCRIPTIVE   GEOMETRY 


127.  To  determine  the  curve  of  intersection 
between  a  plane  and  a  prism. 

Principle,  A  prism  may  be  considered  as 
a  pyramid  with  its  apex  at  infinity;  hence,  this 
problem  in  no  wise  differs  in  principle  and 
method  from  that  of  the  pyramid  (Art/120). 

Construction.  Fig.  165.  Given  the  prism 
the  elements  of  which  are  A^  B^  (7,  and  D 
intersected  by  plane  N.  Points  5,  c,  and  d 
are  the  points  in  which  elements  B^  (7,  and  I), 
respectively,  pierce  plane  iV,  determined  by 
the  use  of  the  vertical  projecting  planes  of  the 
elements  (Art.  59,  page  42).  Element  A,  if 
extended,  will  in  like  manner  be  found  to 
pierce  plane  i^T  at  point  a.  By  connecting  ah, 
be,  ed,  and  da,  the  required  lines  of  intersec- 
tion are  determined.  But,  since  point  a  does 
not  lie  on  the  given  prism,  only  the  portions 
of  lines  ab  and  da  which  lie-  on  the  prism,  i.e. 
eh  and  df,  are  required,  and  plane  iV"  intersects 
the  top  base  of  the  prism  in  line  ef. 

Points  e  and  /  may  be  obtained  by  the  use 
of  the  vertical  projecting  planes  of  lines  ^ 
and  F^  in  which  case  point  a  is  not  needed. 


Fig.  1 65. 


THE  HELICAL  CONVOLUTE 


93 


The  construction  may  be  shortened  by 
observing  that  since  the  vertical  projecting 
planes  of  the  elements  are  parallel,  their  lines 
of  intersection  with  the  given  plane  are 
parallel. 

The  true  size  of  the  cut  section  is  deter- 
mined by  revolving  each  of  its  points  into  H 
about  HN  A^  an  axis. 

128.  To  develop  the  prism. 

Method.  Determine  the  true  size  of  a 
right  section  and  proceed  as  in  the  case  of 
cylinder  (Art.  125,  page  88),  or  revolve  the 
prism  parallel  to  a  coordinate  plane  and  pro- 
ceed as  for  the  cylinder  (Art.  126,  page  91). 

129.  The  helical  convolute. 

Fig.  166  illustrates  a  plane  triangle  tangent 
to  a  right-circular  cylinder,  the  base  of  the 
triangle  being  equal  to  the  circumference  of 
the  base  of  the  cylinder.  If  the  triangular 
surface  be  wrapped  about  the  cylinder,  the 
point  e  will  come  in  contact  with  the  cylinder 
at  Cj,  a  at  a^,  and  the  hypotenuse  ac  will  be- 
come a  helix  having  a^c^  for  the  pitch,  and  the 
angle  acd   will   be   the  pitch   angle.     If   the 


right  line  ac  be  revolved  about  the  cylinder 
while  remaining  in  contact  with,  and  tangent 
to,  the  helix,*  and  making  the  constant  angle 
6  with  the  plane  of  the  base,  it  will  generate  a 
convolute  of  two  nappes.  The  lower  nappe, 
which  is  generated  by  the  variable  portion  of 
the  line  below  the  point  of  tangency,  will 
alone  be  considered. 


Fig.  166. 


»  For  the  theory  and  construction  of  the  helix  and 
involute  curves  see  page  104  of  "Elements  of  Mechanical 
Drawing"  of  this  series. 


94 


DESCRIPTIVE   GEOMETRY 


130.    To  draw  elements  of  the  surface  of  the 
helical  convolute. 

Construction.      Fig.    167.     Let    the  di- 
ameter, aH^^   and    tlie   pitch,  aV,  of   the  re- 


quired helical  convolute  be  given.  Then  will 
a''V'h°e°<f  be  the  vertical  projection  of  the  helix, 
and  the  tangents  to  the  space  helix  will  be  the 
elements  of  the  convolute.     From  any  point 


Fig.  168. 

h  draw  a  tangent  to  the  helix.  Its  horizontal 
projection,  hV^  will  be  tangent  to  the  circle 
of  the  base  at  5\  and  the  length  of  this  pro- 
jection will  equal  the  arc  Ve^c^ ;  therefore,  P 


DEVELOPMENT  OF  HELICAL  CONVOLUTE 


95 


will  be  the  horizontal  projection  of  the  hori- 
zontal trace  of  the.  element  hk,  the  vertical 
projection  of  which  will  be  Ifk".  Similarly 
traces  of  other  elements  may  be  found,  and 
their  locus  will  be  points  in  the  base  of  the 
convolute,  which  curve  is  an  involute  of  the 
base  of  the  cylinder  on  which  the  helix  is 
described.  Any  element  of  the  convolute  may 
now  be  obtained  by  drawing  a  tangent  to  the 
circle  of  the  base  of  the  cylinder,  and  limited 
by  the  involute  of  this  same  circle. 

131.    To  develop  the  helical  convolute. 

Since  this  surface  is  developable  (Art.  98, 
page  67),  it  may  be  rolled  upon  a  plane  on 
which  the  elements  will  appear  in  their  true 
lengths  as  tangents  to  the  developed  helix, 
which,  being  a  curve  of  constant  curvature, 
will  be  a  circle.  The  developed  surface  will 
be  an  area  bounded  by  this  circle  and  its 
involute. 

The  radius  of  the  circle  of  the  developed 
helix  is  determined  in  the  following  manner: 
In  Fig.  168  a,  h,  and  e  are  points  in  the  helix, 
and  ht  is  a  tangent  to  the  helix  at  5,  which 
point  is  equally  distant  from  a  and  c.     The 


projection  of  these  points  on  to  the  plane  of 
the  right  section  of  the  cylinder  passed  through 
h  will  be  at  e  and  d.  ac  is  the  chord  of  a  cir- 
cular arc  drawn  through  a,  6,  and  <?,  approxi- 
mating the  curvature  of  the  helix,  hi  is  the 
diameter  of  this  circular  arc,  and  Ich  a  triangle 
inscribed  in  its  semicircumference,  and,  there- 
fore, a  right  triangle.  Similarly  the  triangle 
hdk^  in  the  plane  of  the  right  section,  is  a  right 
triangle.  In  the  triangles  hcl  and  hdk^  he  will 
be  a  mean  proportional  between  fh  and  5Z,  and 
hd  will  be  a  mean  proportional  between /5  and 
hk.  Substituting  R  for  the  radius  of  curva- 
ture of  the  helix,  — ,  and  r  for  the  radius  of 
the  circle  which  is  the  projection  of  the  helix, 
— ^,  we  may  obtain  he  =  fh  x  2  R  and  hd:  =fh 
X  2r.  Dividing  the  second  by  the  first, 
rz^  —  — ,  but  —  =  cos  8.  when  8  is  the  angle 

which  the  chord  he  makes  with  the  horizontal 
— — -.     As  points  a  and  c 

COS^yS 


plane;  hence,  li  = 


96 


DESCRIPTIVE   GEOMETRY 


appi'oacli  each  other,  the  chord  he  will  ap- 
proach ht^  the  tangent  to  the  helix  at  6,  and 
at  the  limit  the  angle  ^  will  equal  ^,  the  angle 
which  the  tangent  niakes  with  the  horizontal 

plane,  so  that  R  =  — r-:. 
cos-*  V 

This  value  may  be  graphically  obtained  in 
the  following  simple  manner:  Fig.  169.  Draw 
a  tangent  to  the  helix  at  h.  From  its  inter- 
section with  the  contour  element  of  the  cyl- 
inder, at  <?,  draw  a  horizontal  line  terminated 
by  the  perpendicular  to  the  tangent  through 
h.     Then  dc  will  be  the  required  radius  for 

the  developed  helix,  since  be  — 


cos^ 


,  and  he  —  cd 


cos  d\  hence,  cc?  = 


ae 


cos'^  6     cos^  6 


Fig.  169. 


Having  determined  the  radius  of  curvature 
of  the  helix,  draw  the  circle  and  its  involute 
to  obtain  the  development  of  the  convolute. 
Fig.  170  is  the  development  of  the  helical  con- 
volute shown  by  its  projections  in  Fig.  167. 


132.  To  determine  the  curve  of  intersection 
between  a  plane  and  a  surface  of  revolution. 

In  the  following  cases  the  axes  of  revolution 
are  considered  to  be  perpendicular  to  one  of 
the  coordinate  planes. 


INTERSECTION  OF  PLANE  WITH  SURFACE  OF  REVOLUTION 


97 


Method.  1.  Pass  a  series  of  auxiliary  cut- 
ting planes  perpendicular  to  the  axis  of  revo- 
lution. These  planes  will  cut  the  surface  in 
circles,  and  the  given  plane  in  right  lines.  2. 
The  intersections  of  the  circles  and  right  lines 
are  points  in  the  required  curve  of  intersection. 

Construction  1.  Fig.  171.  Given  the 
ellipsoid  with  its  axis  perpendicular  to  V 
intersected  by  the  plane  *S'.  Plane  Y  is  one 
of  a  series  of  auxiliary  planes  perpendicular  to 
the  axis  of  the  ellipsoid.  Then  HY  is  parallel 
to  GL;  it  cuts  from  the  ellipsoid  a  circle  which, 
in  horizontal  projection,  coincides  with  KY, 
and  in  vertical  projection  is  a  circle,  and  it  cuts 
from  the  given  plane  S  the  line  B,  intersect- 
ing the  circle  at  t  and  g,  two  points  in  the 
required  curve  of  intersection  between  the 
ellipsoid  and  the  given  plane.  Repeat  this 
process  until  a  sufficient  number  of  points  is 
determined  to  trace  a  smooth  curve. 

It  is  convenient  to  know  at  the  outset  be- 
tween what  limits  the  auxiliary  planes  parallel 
to  H  should  be  passed,  thus  definitely  locating 
the  extremities  of  the  curve.  This  may  be 
accomplished  by  passing   the  meridian  plane 


DESCRIPTIVE  GEOMETRY 


CT,  perpendicular  to  S,  intersecting  S  in  line 
J)  upon  which  the  required  points  a  and  b  are' 
found.  It  will  be  noticed  that  D"  is  an  axis 
of  symmetry  of  the  vertical  projection  of  the 
curve. 

It  is  also  important  that  auxiliary  planes  be 
so  chosen  as  to  determine  points  of  tangency 
with  contour  lines.  Two  such  planes,  which 
should  be  used  in  the  case  of  the  ellipsoid,  are 
the  plane  X,  which  cuts  the  surface  in  its 
greatest  parallel  and  contour  line  in  vertical 
projection,  and  plane  W,  which  cuts  the  sur- 
face in  a  principal  meridian.  The  former  of 
these  determines  the  points  of  tangency,  e"  and 
/",  in  vertical  projection,  and  the  latter  defines 
similar  points  e^  and  c?*  in  horizontal  projection. 
Such  points  as  a,  b,  c,  c?,  e,  and  /  are  designated 
as  critical  points  of  the  curve. 

Construction  2.  Fig.  172.  Given  the 
torus  with  its  axis  perpendicular  to  H  inter- 
sected by  plane  iV.  The  auxiliary  planes  are 
parallel  to  JI,  and  the  critical  points  are  a,  b, 
on  plane  U;  e,f  on  plane  X;  and  c,  d  on  plane 
W.  Also  the  highest  points,  ^,  i,  and  the  low- 
est points,  Z,  k,  should  be  determined. 


CHAPTER   VI 


INTERSECTION  OF  SURFACES 


133.  Whenever  the  surfaces  of  two  bodies 
intersect,  it  becomes  necessary  to  determine 
the  line  of  their  intersection  in  order  to  illus- 
trate and  develop  the  surfaces.  The  charac- 
ter of  these  lines,  which  are  common  to  both 
surfaces,  is  determined  bv  the  nature  of  the 
surfaces,  and  by  their  relative  size  and  posi- 
tion. The  principles  involved  in  the  determi- 
nation of  these  lines,  their  projections,  and 
the  development  of  the  intersecting  surfaces, 
are  fully  treated  in  the  chapter  on  the  inters 
section  of  planes  and  surfaces;  but  it  is  neces- 
sary to  consider  the  character  of  the  auxiliary 
cutting  planes  and  the  methods  of  using  them 


in  order  to  cut  elements  from  two  surfaces 
instead  of  one. 

134.  Character  of  Auxiliary  Cutting  Surfaces. 
The  auxiliary  cutting  surfaces  have  been  re- 
ferred to  as  planes,  but  cylinders  and  spheres 
are  also  used  whenever  they  will  serve  to  cut 
the  intersecting  surfaces  in  right  lines  or 
circular  arcs. 

The  character  of  the  auxiliary  plane,  or  sur- 
face, and  tlie  method  of  using  it,  is  dependent 
upon  the  nature  of  the  intersecting  surfaces, 
since  it  is  most  desirable,  and  generally  possi- 
ble, to  cut  lines  from  the  surfaces,  which  shall 
be  either  right  lines  or  circular  arcs. 


99 


100 


DESCRIPTIVE  GEOMETRY 


The  following  cases  illustrate  the  influence 
of  the  type  of  the  intersecting  surfaces  on  the 
character  and  position  of  the  auxiliary  cutting 
planes  or  surfaces : 

Case  1.  Cylinder  and  cone  with  axes 
oblique  to  the  coordinate  plane  :  Use  auxiliary 
planes  containing  the  apex  of  the  cone  and 
parallel  to  the  axis  of  the  cylinder. 

Case  2.  Two  cylinders  with  axes  oblique 
to  the  coordinate  plane  :  Use  auxiliary  planes 
parallel  to  the  axes  of  the  cylinders. 

Case  3.  Cone  and  cone  with  axes  oblique 
to  the  coordinate  planes:  Use  auxiliary  planes 
containing  the  apices  of  the  cones. 

Case  4.  A  single  and  a  double  curved  sur- 
face of  revolution  with  parallel  axes  wliich  are 
perpendicular  to  a  coordinate  plane  :  Use 
auxiliary  planes  perpendicular  to  the  axes. 

Case  5.  A  single  and  a  double  curved  sur- 
face of  revolution,  two  single-curved  surfaces 
of  revolution,  or  two  double-curved  surfaces 
of  revolution,  with  axes  oblique  to  each  other 
but  intersecting,  and  parallel  to  one  of  the 
coordinate    planes  :     Use     auxiliary    cutting 


spheres  with  centers  at  the  intersection  of  the 
axes. 

.  Case  6.  A  double-curved  surface  of  revo- 
lution with  axis  perpendicular  to  a  coordinate 
plane,  and  any  single-curved  surface :  If  the 
single-curved  surface  be  a  cylinder,  use  auxil- 
iary cutting  C3dinders  with  axes  parallel  to 
that  of  the  cylinder,  intersecting  the  axis  of 
the  double-curved  surface,  and  cutting  circles 
therefrom.  If  the  single-curved  surface  be  a 
cone,  use  auxiliary  cones  with  apices  common 
with  that  of  the  given  cone  and  cutting  the 
double-curved  surface  in  circles. 

Case  7.  A  prism  may  be  substituted  for 
the  cylinder,  or  a  pyramid  for  the  cone,  in 
each  of  the  above  cases,  save  Case  6. 

Case  8.  Prisms  and  pyramids,  two  of  the 
same  or  opposite  kind :  Auxiliary  planes  not 
required.  Determine  the  intersection  of  the 
.edges  of  each  with  the  faces  of  the  other. 

135.  To  determine  the  curve  of  intersection 
between  a  cone  and  cylinder  with  axes  oblique 
to  the  coordinate  plane.  * 

*  It  is  customary  to  represent  these  surfaces  with  one  or 
both  of  the  bases  resting  on  a  coordinate  plane. 


INTERSECTION  OF  CONE  AND  CYLINDER. 


i\^iA    101 


Principle.  Since  the  curve  of  intersection 
must  be  a  line  common  to  both  surfaces,  it 
will  be  drawn  through  the  points  common  to 
intersecting  elements.  A  series  of  cutting 
planes  passed  through  the  apex  of  the  cone 
and  parallel  to  the  axis  of  the  cylinder  w^ill 
cut  elements  from  both  the  cone  and  the  cyl- 
inder, and  since  these  elements  lie  in  the  same 
plane,  tliey  will  intersect,  their  point  of  inter- 
section being  common  to  both  surfaces,  and 
therefore  a  point  in  the  required  curve. 

Method.  1.  Draw  a  line  through  the 
apex  of  the  cone  and  parallel  to  the  axis  of 
the  cylinder.  2.  Determine  its  trace  in  the 
plane  of  the  bases  of  the  cylinder  and  cone. 
3.  Through  this  trace  draw  lines  in  the 
plane  of  the  bases  of  cylinder  and  cone  cutting 
these  bases.  These  lines  will  be  the  traces  of 
the  auxiliary  cutting  planes.  4.  From  the 
points  of  intersection  of  these  traces  with  the 
bases  of  cone  and  cylinder  draw  the  elements 
of  these  surfaces.  5.  Draw  the  required  curve 
of  intersection  through  the  points  of  intersec- 
tion of  the  elements  of  cylinder  and  cone. 


Fig.  173. 


102 


DESCRIPTIVE   GEOMETRY 


Construction.  Fig.  174.  Through  the 
apex  of  the  cone  draw  line  A  parallel  to  the 
axis,  or  elements,  of  the  cylinder.  This  line 
will  be  common  to  airauxiliary  cutting  planes, 
and  its  horizontal  trace,  b'',  will  be  a  point 
common  to  all  their  horizontal  traces.  IfN'm 
one  such  trace  which  cuts,  or  is  tangent  to, 
the  base  of  the  cylinder  at  c^*,  and  cuts  the 
base  of  the  cone  in  d'^  and  e''.  Since  these  are 
points  in  elements  cut  from  the  cylinder  and 
cone  by  the  auxiliary  plane  iV,  the  horizontal 
and  vertical  projections  of  the  elements  may 
be  drawn.  Line  U  will  be  the  element  cut 
from  the  cylinder,  while  da  and  ea  are  the 
elements  cut  from  the  cone.  The  intersection 
of  these  elements  at/ and ^  will  be  points  com- 
mon to  the  cylinder  and  cone ;  hence,  points 
in  the  required  curve  of  intersection. 

The  planes  iV  and  *S'  are  tangent  planes,  the 
former  to  the  cylinder  and  the  latter  to  the 
cone ;  therefore,  they  will  determine  but  two 
points  each.  Intermediate  planes,  such  as  M, 
will  cut  four  elements  each,  two  from  each 
surface,  and  will  determine  four  points  of  in- 
tersection.    Draw  as  many  such  planes  as  may 


INTERSECTION  OF  CONE   AND  CYLINDER 


103 


be  necessary  to  determine  the  required  curve. 
Since  it  is  desirable  to  determine  the  points  of 
tangency  between  the  curve  and  contour 
elements,  strive  to  locate  the  intermediate 
planes  so  that  all  such  elements  may  be  cut  by 
the*  auxiliary  planes. 

136.  Order  and  Choice  of  Cutting  Planes.  The 
tangent  planes  should  be  drawn  first,  thus 
determining  limiting  points  in  the  curve,  such 
as/,  g^  k,  I,  and,  as  will  be  shown  in  Art.  137, 
determining  whether  there  be  one  or  two 
curves  of  intersection.  Xext  pass  planes  cut- 
ting contour  elements  in  both  views.  Jf  is 
one  such  plane  cutting  a  contour  element  in 
the  horizontal  projection  of  the  cylinder  and 
determining  points  i  and  Z,  limiting  points  in 
the  horizontal  projection  of  the  curve. 

137.  To  determine  if  there  be  one  or  two 
curves  of  intersection.  It  is  possible  to  deter- 
mine the  number  of  curves  of  intersection 
between  the  cylinder  and  cone  previous  to 
finding  the  required  points  in  the  line,  or  lines, 
of  intersection.  In  Fig.  174  it  will  be  observed 
that   there   is   but   one   continuous   curve   of 


intersection,  and  this  is  due  to  the  fact  that 
only  one  of  the  two  tangent  auxiliary  planes 
that  might  be  drawn  to  the  cj'linder  will  cut 
the  cone.  If  the  surfaces  had  been  of  such 
size,  or  so  situated,  that  the  two  planes  tan- 
gent to  the  cylinder  had  cut  the  cone,  as  in- 
dicated by  Fig.  175,  in  which  only  the  traces 
of  the  cylinder,  cone,  and  planes  are  drawn, 


there  would  have  been  two  curves,  the  cylinder 
passing  directly  through  the  cone.  Again  if 
two  planes  tangent  to  the  cone  had  cut  the 
cylinder,  as  in  Fig.  176,  the  cone  would  have 
pierced  the  cylinder,  making  two  independent 
curves  of  intersection.  The  condition  shown 
by  Fig.  177  is  that  of  the  problem  solved, 
and  indicates  but  one  curve   of   intersection. 


104 


DESCRIPTIVE   GEOMETRY 


138.  To  determine  the  visible  portions  of  the 
curve.  1.  A  point  in  11  curve  of  intersection 
is  visible  only  when  it  lies  at  the  intersection 
of  two  visible  elemeNts.  2.  The  curve  of 
intersection  being  common  to  both  surfaces 
will  be  visible  in  vertical  projection  only  as 
far  as  it  lies  on  the  front  portion  of  both  sur- 
faces. 3.  The  horizontal  projection  of  the 
curve  is  visible  only  as  far  as  it  lies  on  the 
upper  portion  of  both  surfaces.  4.  The  point 
of  passing  from  visibility  to  invisibility  is  al- 
ways on  a  contour  element  of  one  of  the  sur- 
faces. Fig.  173  represents  the  cone  and 
cylinder  with  only  the  visible  portions  shown. 

139.  To  determine  the  curve  of  intersection 
between  two  cylinders,  the  axes  of  which  are 
oblique  to  the  coordinate  planes. 

Principle.  Auxiliary  planes  parallel  to 
the  axes  of  both  cylinders  will  cut  elements 
from  each,  and  their  intersections  will  deter- 
mine points  in  the  required  curve.  Since  the 
auxiliary  planes  are  parallel  to  the  axes,  they 
will  be  parallel  to  each  other,  and  their  traces 
will  be  parallel. 


Method.  1.  Through  one  of  the  axes  pass 
a  plane  parallel  to  the  other  axis  (Art.  74, 
page  52),  and  this  will  be  one  of  the  auxiliary " 
planes  to  which  the  other  cutting  planes  will 
be  parallel.  2.  Determine  if  there  be  one  or 
two  curves  of  intersection.  8.  Beginning 
with  one  of  tire  tangent  planes  pass  auxiliary 
planes  through  the  contour  elements  of  each 
view,  using  such  additional  planes  as  may  be 
necessary  to  determine  tJie  requisite  number 
of  points  in  the  curve.  4.  Draw  the  curve, 
determining  the  visible  portions  by  Art.  138, 
page  105. 

140.  To  determine  the  curve  of  intersection 
between  two  cones,  the  axes  of  which  are 
oblique  to  the  coordinate  planes. 

Pkinciple.  Auxiliary  planes  which  contain 
the  apices  of  the  cones  will  cut  elements  from 
each  of  the  surfaces.  Hence,  all  the  cutting 
planes  will  contain  the  line  joining  the  apices 
of  the  cones,  and  all  the  traces  of  the  auxiliary 
planes  will  intersect  in  the  traces  of  this  line. 

The  solution  of  this  problem  is  similar  to 
that  of  Art.  135,  page  100. 


INTERSECTION  OF  ELLIPSOID   AND  CYLINDER 


105 


141.  To  determine  the  curve  of  intersection 
between  an  ellipsoid  and  an  oblique  cylinder. 

Principle.  Auxiliary  cylinders  with  axes 
parallel  to  that  of  the  oblique  cylinder  and 
intersecting  the  axis  of  the  ellipsoid  may  be 
chosen  of  such  section  as  to  cut  circles  from 
the  ellipsoid  and  elements  from  the  C3linder. 

Construction.  Fig.  178.  Draw  any 
parallel  h^cf  in  vertical  projection  and  con- 
ceive it  to  be  the  horizontal  section  of  an 
auxiliary  cylinder,  the  axis  of  which  is  de. 
This  auxiliary  cylinder  will  intersect  the 
horizontal  coordinate  plane  in  a  circle  having 
e*  for  its  center  and  a  diameter  equal  to  h'c"^ 
since  all  horizontal  sections  will  be  equal. 
Points  /  and  k  lie  at  the  intersection  of  the 
bases  of  the  given  and  auxiliary  cylinder,  and, 
therefore,  23oints  in  elements  common  to  both 
cylinders.  The  intersections  of  these  elements 
with  the  parallel  cut  from  the  ellipsoid,  I  and 
0,  will  be  points  common  to  the  two  surfaces 
and,  therefore,  points  in  the  required  curve. 
Similarly  determine  the  necessary  number  of 
points  for  the  drawing  of  a  smooth  curve. 


106 


DESCRIPTIVE  GEOMETRY 


142.  To  determiile  the  curve  of  intersection 
between  a  torus  and  a  cylinder,  the  axes  of 
which  are  perpendicular  to  the  horizontal  coor- 
dinate plane. 

Principle.  Auxiliary  planes  perpendicular 
to  the  axes  will  cut  each  of  the  surfaces  in 
circles  the  intersections  of  which  will  deter- 
mine points  common  to  both  surfaces;  or, 
meridian  planes  of  the  torus  will  cut  elements 
from  the  cylinder  and  meridians  from  the 
torus,  the  intersections  of  which  will  be  points 
in  the  required  curve. 

Method.  1.  Determine  the  lowest  points 
in  the  curve  on  the  inner  and  outer  surfaces 
of  the  torus  by  a  meridian  cutting  plane  con- 
taining the  axis  of  the  cylinder.  2.  Deter- 
mine the  points  of  tangency  with  the  contour 
elements  of  the  cylinder  by  meridian  cutting 
planes  containing  said  elements.  3.  Deter- 
mine the  highest  points  in  the  curve  by  an 
auxiliary  plane  cutting  the  highest  parallel. 
4.  Determine  intermediate  points  by  auxiliary 
planes  cutting  parallels  from  the  torus  and 
circles  from  the  cylinder. 

Construction.     Fig.  179.     Through   the 


axis  of  the  cylinder  pass  the  meridian  plane  of 
the  torus,  cutting  elements  from  the  cylinder 
and  a  meridian  from  the  torus.  Revolve  this 
plane  about  the  axis  of  the  torus  until  it  is 
parallel  to  V.  In  this  position  the  vertical 
projections  of  the  elements  will  be  Al  and  B^ 
and  their  intersections  with  the  circle  cut  from 
the  torus  will  be  at  cj  and  c?J.  In  counter- 
revolution these  points  will  fall  at  (f  and  c?*, 
thus  determining  the  lowest  points  of  the 
curve  on  the  inner  and  outer  surface  of  the 
torus.  The  meridian  planes  N  and  M  will  cut 
contour  elements  from  the  cylinder,  and  the 
vertical  projections  of  their  points  of  intersec- 
tion with  the  torus  at  e°  and/'  will  be  deter- 
mined as  in  the  previous  case.  In  this  manner 
all  the  points  of  intersection  may  be  found  ;  or, 
planes  cutting  parallels  from  the  torus  may  be 
used,  as  H  and  S,  the  former  of  which  is  tan- 
gent to  the  upper  surface  of  the  torus  cutting 
it  and  the  cylinder  in  circles  which  intersect 
at  points  k  and  I.  The  remaining  points 
necessary  to  the  determination  of  the  curve 
may  be  similarly  found  as  shown  by  the  auxil- 
iary j^lane  ^S*. 


INTERSECTION  OF  TORUS  AND  (Ti'LINDER 


107 


Fig    179. 


108 


DESCRIPTIVE   GEOMETRY 


143.  To  determine  the  curve  of  intersection 
between  an  ellipsoid  and  a  paraboloid,  the  axes 
of  which  intersect  and  are  parallel  to  the  ver- 
tical coordinate  planes.  , 

Principle.  Auxiliary  spheres  having  their 
centers  at  the  intersection  of  the  axes  of  the 
surfaces  of  revolution  will  cut  the  intersecting 
surfaces  in  circles,  one  projection  of  which  will 
be  right  lines. 

Construction.  Fig.  180  illustrates  this 
case.  It  will  be  observed  that  the  horizontal 
projection  of  the  parabola  is  omitted  since  the 
curve  of  intersection  is  completely  determined 
by  the  vertical  projections  of  the  two  surfaces 
and  the  horizontal  projections  of  the  parallels 
of  the  ellipsoid. 


OF    THE  ^ 

UNIVER^^TY    I 


CHAPTER   VII 


WARPED  SURFACES 


144.  Warped  Surfaces*  are  ruled  surfaces, 
being  generated  by  the  motion  of  a  right  line, 
the  consecutive  positions  of  which  do  not  Ue 
in  the  same  plane.  The  right-line  generatrix 
is  governed  in  two  distinct  ways  : 

1.  By  contact  with  three  linear  directrices. 

2.  By  contact  with  two  linear  directrices 
while  maintaining  parallelism  with  a  plane 
or  other  type  of  surface.  If  the  governing 
surface  is  a  plane,  it  is  called  a  plane  director; 
if  a  cone,  it  is  called  a  cone  director,  and  the 
generatrix  must  always  be  parallel  to  one  of 
its  elements. 

*  The  classification  of  surfaces  considered  in  Chapter 
III,  and  especially  that  portion  of  the  subject  relating  to 
warped  surfaces.  Arts.  100  and  101,  page  68,  should  be 
reviewed  previous  to  studying  this  chapter. 


The  following  types  will  be  considered,  the 
first  two  being  employed  to  set  forth  the  char- 
acteristic features  of  this  class  of  surfaces: 

A  surface  having  its  generatrix  governed 
by  three  curvilinear  directrices. 

A  surface  liaving  its  generatrix  governed 
by  two  curvilinear  directrices  and  a  plane 
director. 

An  hyperbolic  paraboloid,  illustrating  a  sur- 
face the  generatrix  of  which  is  governed  by 
two  rectilinear  directrices  and  a  plane  director. 

The  oblique  helicoid,  illustrating  a  surface 
the  generatrix  of  which  is  governed  by  two 
curvilinear  directrices  and  a  cone  director. 

The  hyperboloid  of  revolution  of  one  nappe, 
illustrating  a  surface  which  may  be  generated 
by  several  methods,  and  the  only  warped  sur- 
face which  is  a  surface  of  revolution. 


109 


no 


DESCRIPTIVE  GEOMETRY 


145.  Having  given  three  curvilinear  direc- 
trices and  a  point  on  one  of  them,  it  is  required 
to  determine  the  two  projections  of  the  element 
of  the  warped  surface  passing  through  the  given 
point. 

Principle.  Right  lines  drawn  from  a  given 
point  on  one  directrix  to  assumed  points  on 
a  second  directrix  will  be  elements  of  a  coni- 
cal surface.  If  the  intersection  between  this 
conical  surface  and  a  third  directrix  be  ob- 
tained, it  will  be  a  point  of  an  element  of  the 
auxiliary  cone,  which  is  also  an  element  of  the 
warped  surface,  since  it  will  be  in  contact  with 
the  three  directrices. 

Method.  1.  Assume  points  on  one  direc- 
trix and  draw  elements  of  an  auxiliary  cone 
to  the  given  point.  2.  Determine  the  inter- 
section between  this  auxiliary  cone  and  the 
third  directrix.  3.  Through  the  given  point 
and  the  point  of  intersection  of  the  curve  and 
directrix  draw  the  required  element. 

Construction.  Fig.  181.  ^,  JB,  and  Care 
three  curvilinear  directrices  of  a  warped  sur- 
face.    It  is  required  to  draw  an  element  of 


the  surface  passing  through  point  d  on  A. 
Assume  points  on  one  of  the  other  directrices, 
in  this  case  C,  and  through  these  points,  e,  /, 
k,  and  ?,  draw  lines  to  d.  Since  (7  is  a  curved 
line,  these  lines  will  be  elements  of  a  cone. 
To  find  the  intersection  between  directrix  B 
and  the  auxiliary  cone,  use  the  auxiliary  cylin- 
der which  horizontally  projects  B.  B'^  will  be 
its  horizontal  trace  and  the  horizontal  projec- 
tion of  the  curve  of  intersection  between  the 
auxiliary  cone  and  cylinder.  Project  the 
horizontal  projections  of  the  points  of  in- 
tersection between  the  elements  of  the  cone 
and  cylinder,  w\  w?*,  r^  and  «*,  to  obtain  points 
in  the  vertical  projection  of  the  curve  of  inter- 
section, w',  m",  r%  and  «',  thereby  determining 
point  0,  the  intersection  of  the  directrix  B  with 
the  auxiliary  cone.  This  point  must  lie  on 
the  auxiliary  cone  since  it  lies  on  the  curve 
of  intersection  between  the  cylinder  and  cone; 
hence,  an  element  drawn  through  o,will  inter- 
sect (7,  the  directrix  of  the  cone,  and  it  will  be 
an  element  of  the  warped  surface  because  it  is 
in  contact  with  the  three  directrices  A,  B,  and 


WARPED  SURFACE 


111 


C;  doj  is,  therefore,  the  required  element  of 
the  warped  surface. 

146.  Having  given  two  curvilinear  directrices 
and  a  plane  director,  to  draw  an  element  of  the 
warped  surface. 

Case  1.  In  which  the  element  is  required 
to  be  drawn  through  a  given  point  on  one  of 
the  directrices. 

Principle.  If  a  plane  be  passed  through 
the  given  point  parallel  to  the  plane  director, 
it  will  cut  the  second  directrix  in  a  point 
which,  if  connected  with  the  given  point, 
will  define  an  element  of  the  surface,  it  being 
parallel  to  the  plane  director  and  in  contact 
with  both  directrices. 

Method.  1.  Through  any  point  in  the 
plane  director  draw  divergent  lines  of  the 
plane.  2.  Through  the  given  point  of  the  di- 
rectrix draw  parallels  to  the  assumed  lines  on 
the  plane  director.  3.  Determine  the  inter- 
section between  the  plane  of  these  lines  and 
the  second  directrix  by  use  of  the  projecting 
cylinder  of  this  directrix.  4.  Connect  this 
point  with  the  given  point. 


112 


DESCRIPTIVE   GEOMETRY 


Construction.  Fig.  182.  A  and  B  are 
the  directrices,  N  the  plane  director,  and  d  the 
given  point.  From  any  point  c,  in  the  plane 
director,  draw  divergent  lines  E,  F,  and  G. 
Through  point  d  draw  dt,  ds^  and  dr  parallel 
to  the  lines  in  the  plane  director,  thus  defin- 
ing a  plane  parallel  to  N.  The  vertical  pro- 
jection of  the  curve  of  intersection  between 
this  plane  and  the  horizontal  projecting  cylin- 
der of  B  will  be  r^s^f.  The  intersection  be- 
tween this  curve  and  B  is  at  m,  and  dm  will 
be  the  required  element  of  the  warped  surface. 

147.  Case  2.  In  which  an  element  is  re- 
quired to  be  drawn  parallel  to  a  line  of  the 
plane  director.  v 

Principle.  If  an  auxiliary  \jylinder  be 
used  which  has  one  of  the  curved  directrices 
for  its  directrix,  and  its  elements  parallel  to 
the  given  line,  it  will  have,  one  element  which 
will  intersect  the  second  directrix.  Such  an 
element  will  be  parallel  to  the  given  line  on 
the  plane  director,  and  in  contact  with  both 
directrices ;  hence,  an  element  of  the  warped 
surface. 


Method.  1.  Through  assumed  points  on 
one  directrix  draw  lines  parallel  to  the  given 
line  in  the  plane  director,  thus  defining  an 
auxiliary  cylinder.  2.  Determine  the  curve 
of  intersection  between  this  cylinder  and  one 
of  the  projecting  cylinders  of  the  second 
directrix.  8.  Through  the  intersection  of 
this  curve  with  the  second  directrix  draw  the 
required  element  parallel  to  the  given  line. 

Construction.  Fig.  183.  A  and  B  are 
the  directrices,  N  the  plane  director,  and  C 
the  given  line  in  the  plane.  e,  /,  k,  and  I 
are  the  assumed  points  on  directrix  A  through 
which  the  elements  of  a  cylinder  are  drawn 
parallel  to  line  O.  This  auxiliary  cylinder 
through  A  will  intersect  the  horizontal  pro- 
jecting cylinder  of  ^  in  a  curve  of  which  S"  is 
the  vertical  projection,  m  is  a  point  common 
to  the  auxiliary  cylinder  and  the  directrix  B^ 
and  dm  ihQ  required  element  of  the  warped 
surface. 

148.  Modifications  of  the  two  types  of  sur- 
faces in  Arts.  145  and  146  may  be  made  to 
include  all  cases  of  warjied  surfaces. 


WARPED  SURFACE 


113 


Fig.    182 


Fig.    183. 


X 


114 


DESCRIPTIVE  GEOMETRY 


In  the  first  type  the  three  linear  directrices 
may  be  curvilinear,  rectilinear,  or  both  curvi- 
linear and  rectilinear. »  In  the  second  type, 
with  two  linear  directrices  and  a  plane  director, 
the  directrices  may  be  curvilinear  or  rectilinear, 
and  a  cone  may  be  substituted  for  the  plane. 
All  conceivable  ruled  surfaces  may  be  gener- 
ated under  one  of  these  conditions. 

149.  The  Hyperbolic  Paraboloid.  If  the  case 
considered  in  Art.  146  be  changed  so  that  the 
two  directrices  be  rectilinear,  while  the  genera- 
trix continues  to  be  governed  by  a  plane  direc- 
tor, the  surface  will  be  an  hyperbolic  paraboloid. 
It  is  so  called  because  cutting  planes  will  in- 
tersect it  in  hyperbolas  or  parabolas.  Figs. 
184  and  185  illustrate  this  surface.  In  Fig. 
184,  A  and  B  are  the  directrices,  and  H  the 
plane  director  of  the  surface.  The  positions 
of  the  generatrix,  or  elements,  are  shown  by 
the  dotted  lines.  As  the  elements  will  divide 
the  directrices  proportionally,  they  may  be 
drawn  by  dividing  the  directrices  into  an 
equal  number  of  parts,  and  connecting  the 
points  in   their  numerical  order. 


This  surface  is  capable  of  a  second  genera- 
tion by  conceiving  the  elements  D  and  0  to  be 
directrices,  and  P  to  be  the  plane  director. 
In  this  case  the  directrices  (7 and  D  are  divided 
proportionally  by  the  elements  which  are  now 
parallel  to  P. 

Again,  we  may  consider  the  lines  A^  E^  and 
B  to  be  three  rectilinear  directrices  governing 
the  motion  of  the  generatrix  J),  which  is 
fully  constrained  and  will  describe  the  same 
surface  as  before ;  but  if  the  three  directrices 
were  not  parallel  to  the  same  plane,  the  char- 
acter of  the  surface  would  be  changed,  and  it 
would  become  an  hyperboloid  of  one  nappe. 

An  interesting  application  of  this  surface  to 
practice  is  found  in  the  pilot,  or  "  cow  catcher," 
of  a  locomotive,  which  consists  of  two  hyper- 
bolic paraboloids  symmetrically  placed  with 
respect  to  a  vertical  plane  through  the  center 
of  the  locomotive.  Figs.  186  and  187  illus- 
trate the  types  which  are  commonly  used.  In 
the  former  the  plane  director  is  vertical  and 
parallel  to  the  rails,  and  in  the  latter  it  is 
horizontal. 


HYPERBOLIC  PARABOLOID 


115 


Fig,  187. 


116 


DESCRIPTIVE   GEOMETRY 


i50-  Through  a  given  point  on  a  directrix,  to 
draw  an  element  of  the  hyperbolic  paraboloid. 

Principle.  TJie  required  element  must  lie 
in  a  plane  containing  the  given  point  and  par- 
allel to  the  plane  director.  A  second  point  in 
this  line  will  lie  at  the  intersection  of  the 
second  directrix  with  the  auxiliary  plane  passed 
through  the  given  point. 

Method.  1.  Through  the  given  point 
draw  two  lines,  each  of  which  is  parallel  to  a 
trace  of  the  plane  director  (Art.  71,  page  50). 
2.  Determine  the  point  of  intersection  of  the 
second  directrix  with  the  plane  of  the  auxiliary 
lines  (Art.  61,  page  44).  3.  Connect  this 
point  of  intersection  with  the  given  point. 

Construction.  Fig.  188.  A  and  B  are  the 
directrices,  and  n  the  given  point.  Through 
n  draw  U  parallel  to  the  horizontal  trace  of 
plane  N,  and  F  parallel  to  the  vertical  trace. 
These  lines  will  determine  a  plane  parallel  to 
iV.  Next  obtain  the  intersection  of  directrix 
B  with  the  plane  of  lines  J5Jand  F.  This  point 
is  m,  and  mn  will  be  the  required  element. 


151.  Having  given  one  projection  of  a  point 
on  an  hyperbolic  paraboloid,  to  determine  the 
other  projection,  and  to  pass  an  element  through 
the  point. 

Construction.  Fig.  1 89.  m*  is  the  hori- 
zontal projection  of  the  given  point  which  lies 
on  the  surface  of  the  hyperbolic  paraboloid 
having  A  and  B  for  its  directrices,  and  iVfor 
the  plane  director.  Through  m  draw  nig  per- 
pendicular to  IT,  and  determine  its  intersection 
with  the  surface,  as  follows  : 

Determine  two  elements,  cd  and  ef^  near  the 
extremities  of  the  directrices  (Art.  150),  the 
work  not  being  shown  in  the  figure.  Divide 
the  portion  of  each  directrix  limited  by  the 
elements  ed  and  ef  into  an  equal  number  of 
parts  to  obtain  elements  of  the  surface  (Art. 
149,  page  114).  Pass  an  auxiliary  plane  X 
through  the  perpendicular  mg.  This  will  in- 
tersect the  elements  at  k,  I,  and  r  ;  the  curve 
*S',  connecting  these  points,  will  be  the  line  of 
intersection  between  the  auxiliary  plane  X  and 
the  warped  surface.     Since  the  curve  S  and 


HYPERBOLIC  PARABOLOID 


117 


the  perpendicular  through  m  lie  in  the  plane 
JT,  their  intersection  will  be  a  point  common 
to  the  perpendicular  and  the  warped  surface. 
Therefore  m*  and  w*  will  be  projections  of  the 
required  point. 

To  obtain  the  required  element,  pass  a  plane 


through  this  point  w,  parallel  to  the  plane 
director  N,  and  determine  its  intersection 
with  one  of  the  directrices  (Art.  loO,  page 
116).  Through  this,  and  the  point  w,  draw 
the  required  element.  The  last  operation  is 
not  illustrated  in  the  figure. 


Fig.  I 


Fig.  189. 


118 


DESCRIPTIVE  GEOMETRY 


152.  Warped  Helicoids.  Suppose  the  line 
5e,  of  Fig.  190,  to  be  revolved  uniformly  about 
the  line  cd  as  an  axis  while  maintaining  the 
angle  0  constant,  and  at  the  same  time  com- 
pelled to  move  in  contact  with,  and  uniformly 
along,  the  axis.  AH  points  in  the  line,  save 
that  one  in  contact  with  the  axis,  will  gener- 
ate helices  of  a  constant  pitch,  and  the  sur- 
face generated  will  be  an  oblique  helicoid. 
The  axis,  and  the  helix  described  by  point  6, 
may  be  considered  as  the  directrices,  and  the 
generatrix  may  be  governed  by  a  cone  which 
is  conaxial  with  the  helicoid.  The  elements 
of  this  cone  will  make  the  angle  0  with  IT. 

The  generatrix  may  also  be  governed  by 
two  helical  directrices  and  a  cone  director,  as 
in  Fig.  191. 

Again,  it  may  be  governed  by  three  direc- 
trices, which  in  this  case.  Fig.  191,  may  be 
the  two  helices  and  the  axis.  The  V-threaded 
screw  is  the  most  familiar  application  of  the 
oblique  helicoid  (Fig.  147,  page  69). 

153.  If  the  generatrix  be  perpendicular  to 
the  axis,  as  in  Fig.  192,  it  may  be  governed 


by  directrices  similar  to  the  preceding,  but 
the  cone  director  will  have  become  a  plane 
director,  and  the  surface  generated  will  be  a 
right  helicoid.  This  type  is  illustrated  by  the 
square-threaded  screw  (Fig.  146,  page  69). 

154.  If  the  generatrix  does  not  intersect 
the  axis,  as  in  the  preceding  cases,  a  more 
general  type  will  be  generated,  as  shown  in 
Fig.  193.  In  this  case  the  generatrix  is  gov- 
erned by  two  helical  directrices  and  a  cone 
director,  the  generatrix  being  tangent  to  the 
cylinder  on  which  the  inner  helix  is  described. 

155.  Hyperboloid  of  Revolution  of  one  Nappe. 
This  is  a  surface  of  revolution  which  may  be 
generated  by  the  revolution  of  an  hyperbola 
about  its  conjugate  axis,  as  illustrated  in  Fig. 
194.  It  is  also  a  warped  surface  in  that  it 
may  be  generated  by  the  revolution  of  a  right 
line  about  an  axis  which  it  does  not  intersect, 
and  to  which  it  is  not  parallel.  Furthermore, 
it  will  be  shown  that  the  rectilinear  generatrix 
may  be  governed  by  three  rectilinear  direc- 
trices, by  three  curvilinear  directrices,  or  by 
two  curvilinear  directrices  and  a  cone  director. 


WARPED  HELICOIDS 


119 


Fig.  191 


Fig.  192. 


Fig.  193. 


120 


DESCRIPTIVE   GEOMETRY 


In  Fig.  194  conceive  the  generatrix  cd  as 
making  the  constant  angle  c^d'h"  with  a  hori- 
zontal plane,  and  revolving  about  a  vertical 
axis  through  o.  Point  h  will  describe  the  circle 
of  the  upper  base  cgl,  point  d  will  describe  the 
circle  of  the  lower  base  dmf,  and  the  point  e, 
the  nearest  to  the  axis,  will  describe  the  circle 
ekn,  which  is  called  the  circle  of  the  gorge. 
All  other  points  of  the  generatrix  will  simi- 
larly describe  circles,  and  by  drawing  these 
parallels  of  the  surface,  the  meridian  line  will 
be  determined,  and  is  an  hyperbola.  Thus 
point  8,  in  the  generatrix  cd,  will  be  in  the 
position  t  when  it  lies  in  the  principal  me- 
ridian plane,  and  f  will  be  a  point  in  the  ver- 
tical contour,  which  is  an  hyperbola. 

156.  Through  any  point  of  the  surface  to 
draw  an  element.  If  one  projection  of  the 
point  be  given,  draw  a  parallel  of  the  surface 
through  this  point  to  determine  the  other  pro- 
jection. Through  the  horizontal  projection 
of  this  point  draw  a  tangent  to  the  circle  of 
the  gorge,  and  it  will  be  the  horizontal  pro- 
jection of  the  required  element,  the  extremi- 


Fig.  194.        m"      f  b" 


HYPERBOLOID  OF  REVOLUTION  OF  ONE  NAPPE 


121 


ties  of  which  lie  in  the  horizontal  projections 
of  the  upper  and  lower  bases  of  the  surface. 
Since  either  extremity  may  be  regarded  as 
lying  in  the  upper  base  of  the  surface,  there 
are  two  tangents  which  may  be  drawn  through 
the  given  point.  They  will  make  equal 
angles  with  the  horizontal  coordinate  plane, 
and  intersect  at  the  circle  of  the  gorge. 

157.  The  Generatrix  may  be  governed  by 
Three  Rectilinear  Directrices.  Fig.  194.  If  two 
elements,  ab  and  erf,  be  drawn  through  point  e 
of  the  circle  of  the  gorge,  either  may  be  taken 
as  the  generatrix  of  the  surface.  One  is 
known  as  an  element  of  the  first  generation, 
and  the  other  as  an  element  of  the  second 
generation. 

Conceive  ah  as  fixed  and  cd  as  the  genera- 
trix. In  the  revolution  about  the  axis,  cd 
will  at  all  times  intersect  aft,  if  these  lines  be 
extended  indefinitely.  This  may  be  proved 
as  follows  :  If  cd  be  in  the  position  indicated 
by  gf^  then  the  horizontal  projections  of  ab 
and  df  will  intersect  in  r*.  This  point  will 
be  equally  distant  from  the  points  of  tangency 


e*  and  Ar*,  and  since  ab  and  gf  make  equal 
angles  with  H,  the  distances  er  and  kr  must 
be  equal,  and  hence,  r  must  be  at  the  inter- 
section of  ab  and  gf.  If,  then,  we  conceive 
three  elements  of  the  surface,  such  as  cd,  gf, 
and  Im,  and  if  we  conceive  ab  as  the  genera- 
trix, it  will  intersect  each  of  these  elements 
and  they  may  be  used  as  directrices. 

Again,  if  three  parallels  be  the  directrices, 
the  generatrix  will  be  fully  constrained. 

158.  The  Generatrix  may  be  governed  by 
Two  Curvilinear  Directrices  and  a  Cone  Director. 
Fig.  194.  Since  the  elements  of  the  surface 
are  parallel  to  the  elements  of  a  cone,  having 
the  angle  d'^e^'b'  for  the  apex  angle,  this  may 
be  used  as  a  cone  director,  the  generatrix  be- 
ing also  governed  by  two  parallels  of  the  sur- 
face, such  as  the  bases,  or  a  base  and  the  circle 
of  the  gorge. 

159.  The  tangent  plane  to  any  point  of  the 
surface  is  determined  by  the  elements  of  the 
two  generations  drawn  through  this  point. 
The  plane  determined  by  lines  ah  and  gf  will 
be  tangent  to  the  surface  at  point  r.  Fig.  194. 


122 


DESCRIPTIVE  GEOMETRY 


1 60.  Through  a  right  line  to  pass  a  plane  tan- 
gent to  any  double-curved  surface  of  revolution. 

By  the  use  of  the  hyperboloid  of  revolution 
of  one  nappe  as  an  auxiliary  surface,  it  is  pos- 
sible-to  make  a  general  solution  of  problems 
requiring  the  determination  of  tangent  planes 
to  double  curved  surfaces  of  revolution,  as 
follows : 

Principle.  If  the  given  right  line  be  re- 
volved about  the  axis  of  the  double-curved 
surface  of  revolution,  it  will  generate  an  hy- 
perboloid of  revolution.  A  plane  tangent  to 
both  surfaces  and  containing  the  given  line, 
which  is  an  element  of  one  of  them,  will  be 
the  required  plane.  Since  one,  and  only  one 
meridian  plane  at  a  point  of  tangency  will  be 
perpendicular  to  the  tangent  plane,  and  as  the 
surfaces  of  revolution  have  a  common  axis,  it 
follows  that  one  meridian  plane  will  cut  a 
line  from  the  tangent  plane  which  will  pass 
through  the  points  of  tangency  on  both  sur- 
faces and  be  tangent  to  both  meridian  curves. 
This  line  and  the  given  line  will  determine 
the  tangent  plane. 


HYPERBOLOID  OF  REVOLUTION  OF  ONE  NAPPE 


123 


Method.  1.  Draw  the  principal  meridian 
section  of  the  hyperboloid  of  revolution 
which  has  the  given  line  for  its  generatrix. 
2.  Draw  a  tangent  to  the  principal  meridian 
sections  of  both  surfaces.  3.  Revolve  this 
line  about  the  axis  of  the  surfaces  until  it 
intersects  the  given  line,  observing  that  its 
point  of  taugency  with  the  hyperbola  is  a 
point  of  the  given  line.  4.  Determine  the 
plane  of  this  tangent  and  the  given  line. 

Construction.  Fig.  195.  Having  de- 
scribed the  hyperboloid  of  revolution  with 
the  given  line  A  as  its  generatrix,  draw  cjfij 


tangent  to  the  meridian  curves.  It  will  be 
the  vertical  projection  of  the  revolved  position 
of  a  line  tangent  to  both  surfaces.  In  counter- 
revolution this  line  will  intersect  the  given 
line  A  at  <?,  which  is  the  counter-revolved 
position  of  the  point  of  tangenc}'  Cy  This 
must  be  so,  since  line  A  is  an  element  of  the 
hyperboloid  of  revolution  and  must  be  in  con- 
tact with  the  parallel  through  Cy  Point  5j 
in  counter-revolution  is  at  J,  and  be  will  be 
a  line  of  the  tangent  plane.  iV  will  be  the 
plane  of  be  and  the  given  line  A,  and,  there- 
fore, the  required  tangent  plane. 


CHAPTER   VIII 
PROBLEMS 


i6i.    Directionsfor  solving  the  Problems. 

The  problems  are  arranged  in  pairs,  allow- 
ing an  instructor  to  assign  them  alternately, 
inasmuch  as  it  would  not  be  a  wise  expendi- 
ture of  time  for  a  student  to  solve  all  of  them. 

The  problems  are  designed  to  be  solved 
within  margin  lines  measuring  7x10  inches, 
one  such  plate  constituting  an  exercise. 

The  notation  of  Art.  6,  page  4,  is  to  be 
used.  The  student  should  remember  that  the 
correct  lettering  of  every  point  and  line,  and 
the  observance  of  the  character  of  lines,  is 
as  much  a  part  of  the  solution  of  the  problem 
as  is  the  correct  location  of  point  or  line. 

The  following  abbreviations  will  be  used  in 
solving  the  problems : 

V-pr.  signifies  Vertical  Projection. 

Hpr.  signifies  Horizontal  Projection. 


P-pr.  signifies  Profile  Projection. 

V-tr.  signifies  Vertical  Trace. 

Il-tr.  signifies  Horizontal  Trace. 

P-tr.  signifies  Profile  Trace. 

The  coordinates  of  points  will  be  designated 
as  follows : 

1st  dimension  is  the  perpendicular  distance 
to  V. 

2nd  dimension  is  tlie  perpendicular  distance 
to^. 

3rd  dimension  is  the  perpendicular  distance 
to  P. 

Distances  above  H  aie  +  ,  and  below  H 
are  — . 

Distances  before  V  are  +,  and  beliind  V 
are  — . 

Measurements  from  P  are  -f-  and  to  the 
left. 


124 


PROBLEMS 


125 


162.    Problems.  The  space  required  for  each  3.    Construct  the  Hpr.  and    V-pr.   of   the 

of   the    first    forty-eight   problems   is    3^  x  5  following  points  (Art.  7,  page  6). 

inches,  and  the  unit  of  measure  is  ^  inch.  a,  6,  2.     J,  5,  —2.     <?,  —  1,  —4.     d,  —4,  3. 

1.    Required  the  distance  from  H^  and  the  e,  4,  0.   /,  —  4,  4.    ^,0,-2.    Z,  4,  —  3.    m,  0,  0. 

Q,  of  each  of  the  following  points  (Art.  7,  w,  4,  —  4. 

4.    Construct  the   S-pr.  and   V-pr.   of   the 

7*'             T"*  following  points  (Art.  7,  page  6). 

V'f  ^9"    I                 :^^  a,  -6,-4.    5,  -4,  4.    c,  5,  4.    ^,  9,  -4. 

!  .             !  e,  -4,  6.     /,   0,   10.      k,   -5,  0.      /,  0,   0. 


page  6). 

,0- 


T^' 


ia* 


+6* 


i6' 


A(/' 


Wl, 


_  O 


6.     w,  8,  8. 


4e' 


^r 


ij'l* 


5.    Fully     describe     the     following     lines 
(Arts.  8,  9,  10,  11,  12,  pages  7  and  8). 


B" 


2.  Required  the  distance  from  V^  and  the 
Q,  of  each  of  tlie  following  points  (Art.  7, 
page  6). 

T6*  T«'e*  t*' 


c* 

- 

f* 

■V. 

-^.- 

f 

T"' 


16- 


I 


ia' 


if 


kk" 


6.    Fully     describe     the     following     lines 
(Arts.  8,  9,  10,  11,  12,  pages  7  and  8). 


9'9^ 


i</* 


/* 

A' 

- 

c 

F' 

c* 

-T 

1 

1 

\n' 
1 

1 
1 

G* 

h' 

^\^ 

•fl* 

N 

H 

Ul 

A* 

«•  / 

^ 

126 


DESCRIPTIVE  GEOMETRY 


7.  Required  the  H-pr.  and  V-pr.  of  the 
following  lines  (Arts.  8,  9,  10,  11,  12,  pages 
7  and  8). 

vl,  inclined  to  F",  inclined  to  H^  in  3  Q. 

B^  parallel  to  H^  inclined  to  F",  in  2  Q. 

(7,  parallel  to  P,  inclined  to  ff  and  F,  in  1  Q. 

D,  perpendicular  to  V,  in  3  Q. 

E,  parallel  to  H,  inclined  to  Fi  in  4  Q. 

F^  inclined  to  F,  lying  in  H^  between  2Q 
and  3  Q. 

8.  Required  the  H-pr.  and  V-pr.  of  the 
following  lines  (Arts.  8,  9,  10,  11,  12,  pages 
7  and  8). 

A.,  inclined  to  F,  inclined  to  H.,  in  1  Q. 

B,  inclined  to  H,  parallel  to  V,  in  3  Q. 

C,  perpendicular  to  F^  in  5  Q. 

D,  inclined  to  F",  parallel  to  ff,  in  2  Q. 

E,  parallel  to  GL,  in  4  Q. 

F.,  inclined  to  H.,  lying  in  F^  between  3Q 
and  4  Q. 

9.  Draw  the  H-pr.  and  V-pr.  of  the  follow- 
ing lines  (Arts.  8  to  15,  pages  7  to  10). 

A  and  B  intersecting  in  5  ^.     A  parallel  to 


F"and  inclined  to  H;  B  inclined  to  Fand  H. 

O  and  I>  intersecting  in  2Q.  C  perpendic- 
ular to  H\  D  parallel  to  GrL. 

E  and  F  not  intersecting.  E  perpendicular 
to  F;  F  inclined  to  Fand  H.     Both  in  4  Q. 

10.  Draw  the  H-pr.  and  V-pr.  of  the  fol- 
lowing lines  (Arts.  8  to  15,  pages  7  to  10). 

A  and  B  parallel,  and  inclined  to  V  and  JT, 
in  5^. 

0  and  D  intersecting  in  1  Q.  C  inclined  to 
Fand  ^;  ^  parallel  to  T^only. 

E  and  F  intersecting  in  3Q.  E  parallel  to 
H  and  inclined  to  V;  F  parallel  to  CrL. 

11.  Required  the  Hpr.^  V-pr.,  and  P-pr. 
of  the  following  lines  (Arts.  21-23,  pages 
14,  15).  State  the  ^'s  in  which  they  appear, 
and  the  direction  of  inclination  (Art.  17, 
page  10). 

a,  —  6,  —  4,  8.  ,  I  c,  6,  4,  9. 

5,  -2,-4,0.         ""^[j,  2, -4,0. 

e,  -  2,  4,  10. 


ah 


(/, -8,  4,  0. 


PROBLEMS 


127 


12.  Required  the  R-pr.^  V-pr.^  and  P-pr. 
of  the  following  lines  (Arts.  21-23,  pages 
14,  15).  State  the  ^'s  in  which  they  appear, 
and  the  direction  of  inclination  (Art.  17, 
page  10). 

a,  —  2,  6,  7.  ,  I  c,  6,  1,  10. 


h,  —  5,  1,  0. 
ef 


«,  -  6,  -  4,  9. 

I/, -2,4,0. 

13.  Required  the  H-pr.   and    V-pr.  of  the 
following  triangles  (Art.  21,  page  14). 

a,  -8,-3,  11.  {d,  -2,  2,8. 

abc\  J,  -  1,  _  10,  7.      def.  e,  -  2,  9,  4. 
c,  0,  -3,0.  |/,-2,  2,  0. 

14.  Required   the  H-pr.  and   V-pr,  of  the 
following  triangles  (Art.  21,  page  14). 

a,  -6,  -6,8.  f(f,  -1,-1,  11. 

ahcl  b,  -1,  -  2,  6.      defi  e,  6,  4,  0. 
e,  0,  -1,0.  [/,  4,  -2i6. 

15.  Three  points,  a,  J,  and  c,  lie  in  P,  and 
in  5^.     a  and  6  have  their  V-prs.  in  the  same 


point,  and  b  and  <?  have  their  ff-prs.  in  the 
same  point,  a  is  4  units  from  V  and  ff;  b 
is  10  units  from  V,  and  c  is  8  units  from  ff. 
Determine  their  H-prs.,  V-prs.  ^  and  P-prs. 
(Arts.  21-23,  pages  14  and  15). 

16.  Three  points,  a,  6,  and  c,  lie  in  P,  and 
in  5  ^.  a  and  ft  have  their  H-prs.  in  the  same 
point,  and  b  and  c  have  their  V-prs.  in  the 
same  point,  a  is  4  units  from  II,  and  8  units 
from  V;  c  is  10  units  from  ff,  and  6  units 
from  V.  Determine  their  H-prs.,  V-prs.,  and 
P-prs.  (Arts.  21—23,  pages  14  and  15). 

17.  Draw  the  H-pr.,  V-pr.,  and  P-pr.  of  line 
ab.  a,  -  2,  12,  0.  b,  9,  -  2,  22.  Determine 
the  prs.  of  the  following  points  in  ab. 

c,  equidistant  from  5^  and  V. 

d,  the  H-tr.  of  the  line.  1      J  Art.  16,  page  10. 

e,  the  V-tr.  of  the  line.  J      [  xA.rt.  24,  page  16. 
/,  the  distance  from  H  twice  that  from  V. 
k,  in  4  Q,  \  units  from  H. 

In  what  ^'s  does  the  line  appear? 

18.  Draw  the  H-pr.,  V-pr.,  and  P-pr.  of  line 
ab.    a,  6,  -  6,  20.    b,  -  2, 11,  0.    Solve  as  for  17. 


128 


DESCRIPTIVE  GEOMETRY 


19.  Make  an  oblique  projection  of  17,  repre- 
senting V,  IT,  and  P  in  their  relative  positions, 
and  the  prs.  of  the  line,  and  points  tliereon. 

20.  Make  an  obliqu«  projection  of  18,  repre- 
senting Fi  -ZT,  and  I*  in  their  relative  positions, 
and  the  prs.  of  the  line,  and  points  thereon. 

21.  Draw  the  prs.  of  line  ab  lying  in  P. 
a,  -  2, 12.  5, 12,  -  4.  Solve  as  for  17,  omit- 
ting c  and  k  (Art.  25,  page  18). 

22.  Draw  the  prs.  of  line  ab  lying  in  P. 
a,  12,  8.  6,  2,  —8.  Solve  as  for  17,  omitting 
c  and  k  (Art.  25,  page  18). 

23.  Draw  the  H-pr.  and  V-pr.  of  lines  A 
and  B,  having  the  following  traces.  Designate 
the  ^'s  in  which  they  appear,  if  produced 
(Art.  26,  page  18).  A,  Il-tr.,  6  units  behind 
V,  and  10  units  to  the  right  of  V-tr.  V-tr., 
8  units  below  IT.  B,  H-tr.,  5  units  before 
V^  and  11  units  to  the  right  of  V-tr.  V-tr.., 
10  units  below  H. 

24.  Draw  the  H-prs.  and  V-prs.  of  lines  C 
and  3,  having  the  following  traces.  Designate 
the    ^'s   in    which   they  appear   if   produced 


(Art.  26,  page  18).  (7,  V-tr.,  4  units  above  IT, 
and  11  units  to  the  right  of  H-tr.  J£-tr.,  10 
units  behind  V.  B,  K-tr.,  9  units  before  V 
and  12  units  to  the  right  of  the  V-tr.  V-tr., 
6  units  above  H. 

25.  Draw  the  H-pr.  and  V-pr.  of  lines  ap- 
pearing in  the  following  ^'s  only  (Art.  24, 
page  16).  A,  1,  4,  3.  B,  1,  2,  3.  Note.  As- 
sume the  traces  of  the  lines  and  proceed  by 
Art.  24,  page  18. 

26.  Solve  as  for  25.     A,  4,  1,  2.     B,  2,  5,  4. 

27.  Solve  as  for  25.     C\  2,  4.  B,  1,  3. 

28.  Solve  as  for  25.      C,  1,3.  B,  4. 

29.  Solve  as  for  25.     E,  4.  F,  Z,  4. 

30.  Solve  as  for  25.     E,  4,  2.         F,  3,  2. 

31.  Solve  as  for  25.     K,  3,  4.         i,  3. 

32.  Solve  as  for  25.     A^  2,  1.         L,  1. 

33.  Draw  the  H-pr.  and  V-pr.  of  line  ab. 
a,  _1,  _6,  14.  b,  -8,  -1,  0.  Through 
the  middle  point  of  ab  draw  line  (7  parallel  to 
V,  and  making  an  angle  of  30°  with  H.  Deter- 
mine the  traces  of  the  plane  of  these  lines 
(Arts.  27-30,  pages  19,  20). 


PROBLEMS 


129 


34.  Draw  Hie  S-pr.  and  V-pr.  of  line  ab. 
a,  -'0,  -  2,  0.  K  - 1,  11,  14.  Through  the 
middle  point  of  ab  draw  line  C  parallel  to  S^ 
and  making  an  angle  of  45°  with  V.  Deter- 
mine the  traces  of  the  plane  of  these  lines 
(Arts.  27-30,  pages  19  and  20). 

35.  Given  line  ab.  a,  3,  3,  10 ;  ft,  7,  9,  0 ; 
and  point  c,  7,  4,  0.  Through  point  c  draw  a 
line  parallel  with  aft,  and  determine  the  traces 
of  the  plane  of  these  lines  (Arts.  27-30,  pages 
19  and  20). 

36.  Given   line    ab.     a,  —  3,  —  8,  9 ;     A,  2, 

—  12,  0  ;  and  point  <?,  —  6,  —  5,  4.  Through 
point  c  draw  a  line  parallel  with  aft,  and  deter- 
mine the  traces  of  the  plane  of  these  lines 
(Arts.  27-30,  pages  19  and  20). 

37.  Given  line  aft.    a,  —  1,  —  6,  14  ;  ft,  —  8, 

—  1,  0 ;  and  point  e,  6,  —  1,  7.  Determine  the 
traces  of  the  plane  of  these  (Art.  33,  page  22). 

38.  Given  line  aft.  a,  —  6,  —  2,  0  ;  ft,  —  1, 
11.  14;  and  point  c,  4,  —  4,  8.  Determine  the 
traces  of  the  plane  of  these  (Art.  33,  page 
22^. 

39.  Determine   the  traces  of  the  plane  in 


which  points  a,  ft,  and  <?  lie.     a,  —  8,  —  2,  22. 
ft,  8,  -  2,  0.  e,  6,  -  9,  14  (Art.  34,  page  22). 

40.  Determine  the  traces  of  the  plane  in 
which  points  a,  ft,  and  c  lie.  a,  —  8,  —  4,  16. 
ft,  -  3,  -  8,  8.  <?,  -  6,  -  13^  0  (Art.  34,  page 
22). 

Note.  In  the  following  problems  the  traces 
of  the  planes  are  parallel  to,  or  make  angles  of 
15°,  or  its  multiple,  with  GL.  This  angle 
may  be  determined  by  inspection. 

41.  Draw  a  triangle  on  plane  M. 
Assume  one  projection  and  proceed 
as  in  Art.  35,  page  22. 

42.  Draw  a  triangle  on  plane  S. 
Assume  one  projection  and  proceed 
as  in  Art.  35,  page  22. 

43.  Determine  the  pr%.  of  a  point 
on  plane  N  which  is  6  units  from  V 
and  H  (Arts.  36,  37,  page  24). 
Through  this  point  draw  three  lines 
on  the  plane  as  follows :  A,  parallel 
to  ff;  B,  parallel  to  V;  C,  oblique 
to  r  and  H  (Arts.  27,  28,  page  10). 


^if 


H8 


VS 


130 


DESCRIPTIVE  GEOMETRY 


44.  Determine  the  prs.  of  a  point  on  plane 
N  which  is  7  units  from  V  and  4  units  from 
jy  (Arts.  36, 37,  page  24).  Through  this  point 
draw  three  lines  on  the  plane  as  follows:  A^ 

parallel  to  F7V;  J5,  parallel  to  HN-, 
C,  through  a  point  on  FTV  12 
units  from  aL  (Arts.  27,  28, 
page  19). 

45.  Determine  the  prs.  of  a  point  on  plane 
S  which  is  6  units  from  V  and  IT  (Arts.  36, 

37,  page   24).     Through   this   point 

9  draw   three    lines   on   the    plane   as 

follows:  A,  passing  through  2  Q  and 

S  Q\   B,  passing  through  3  Q  and  4 

Q  ;    C,   passing    through  2  Q,   3  Q,   and  4  Q 

(Arts.  27,  28,  page  19). 

46.  Determine  the  prs.  of  a  point  on  plane 


*S'  which  is  7  units  from  F'and  4  units  from  H 
(Arts.  36,  37,  page  24).  Through  this  point 
draw   three    lines    on   the   plane   as  ^ 

follows:  J.,  parallel  to  V\  B,  paral-   — -t'^— - 
lei  to  JT;  (7,  passing  through  2  Q,  3   / 
Q,  and  4  Q  (Arts.  27,  28,  page  19). 

47.  Determine   the  prs.    of  the   following 
points  on  plane  iV,  but  not  lying  in 
a  profile  plane.     «,  6  units  from  F'in 

SQ;  6,  4  units  from  H  \n  3  Q  ;    <?,  2 

units  from   V  in  4  Q  (Arts.  36,  37, 
page  24). 

48.  Determine   the  prs.    of    the    following 
points  on  plane  B.,  but  not  lying  in      ^^ 

a  profile  plane,     a,  4  units  from  H     hr     s 

in  3  Q  \  b,  2  units  from  F'in  i  ^  ;  <?,  

4  units  from  R  in  2  Q  (Arts.  36,  37,  page  24). 


VN 


14 


10 


Unit  of  measure,  J  inch.    Space  required  for  each  problem,  2^  x  3  inches.    Measurements  from  GL,  in  light  type, 
and  from  right-hand  division  line,  in  heavy  typa. 


PLATE  1 


16 


6  72 


6* 

-9                          1 

a' 

-6 

12 

a' 

-2 

b" 

'-6 

10 


8     II 


12 


Determine  the  true  length  of  line  A.    Case  1.   (Art.  39,  page  27.)    Case  2.   (Arts.  40,  41,  page  28.) 


Unit  of  measure,  }  Inch.    Space  required  for  each  problem,  2^  x  3  Inches.    Angles  between   GL  and  traces  of 
planes,  multiples  of  15°.    Measurements  from  GL,  In  light  type,  and  from  right-hand  division  line.   In  heavy  type. 


PLATE  2 


HN 


10 


3-^C' 


VN 


HN 


4t-C* 


VN 


Point <;  lies  in  N,  -4,-2 


Point  c  lies  in  N,  -2,-4 


i? 


12 


HN 


2-"C« 


VN 


±/.A 


8-^C 


Point  c  lies  In  plane  N.    Determine  its  distance  from  VN  and  HN.    (Art.  36,  page  24;  Art.  42,  page  30. 


Unit  of  m 
planes,  multipli 


easure,   }  inch.      Space  required  for  each  problem,  2^  x  3  inches.     Angles  between  GL,  and  traces  of      Dl    ATP     "k 
iples  of  15^.    Measurements  from  GL,  in  light  type,  and  from  right-hand  division  line,  in  heavy  type.       •    ^'^  I  t    O 


19 


13  8 


»k  ^3 


10 


16 


5 

'^  ,:■'' 

20        ^^,   12 

VR 

5 

Problems  1-6.    Une  A  lies  in  plane  S.    Determine  its  length  by  revolving  it  into  V  and  into  H,  about  VS  and  HS  as  axes.    (Art. 
43,  page  31.) 

Problems  7-12.    line  B  and  point  e  lie  in  plane  R.    Determine  the  distance  between  them.    (Arts.  42,  43,  pages  SO,  31.) 


Unit  of  measure,  J  Inch.     Space    required    for    each   problem,    2^x3   inches.      Angles   between    GL    and    traces    of      PIATF     4 
planes,  multiples  of  15°.    Measurements  from   GL,   in  light  type,   and  from  right-hand  division  line,  in  heavy  type. 


10  3 


HN 


17         12  10 


19 


\2.^-J ^2 

>I8     Il4'll 


12 


4,< 

C3/ 

131 

8' 

6 

2^-' 

2 

2V 

1    / 

Determine  the  true  size  of  the  polygon  by  revolving  it  into  V  or  H.    (Art.  43,  page  31.) 
Note.  —  In  Problems  7-12  the  traces  of  the  plane  of  the  polygon  should  first  be  determined. 


Unit  of  measure,   |  incb.      Space  required  for    each    problem,    5x7   inches.     Angles    between    GL    and    traces    of 
planes,  multiples  of  15=.      Measurements  from  GL,  in  lig^ht  tjrpe,   and  from  right-hand  division  line,  in  heavy  type. 


PLATE  6 


Draw  the  prs.  of  an  equilat- 
eral triangle  on  plane  N.  Its 
center  is  point  a,  -6,  -4.  One 
side  of  the  triangle  is  7  units 
long  and  parallel  to  V. 


Dra-w  the  prs.  of  a  regular 
hexagon  on  plane  R.  Point  b 
is  one  extremity  of  a  long 
diameter  coinciding  with  line 
A.  Side  of  hexagon  is  5  units 
long. 


Draw  the  prs.  of  a  square 
on  plane  S.  Its  center  is  point 
c,  8,  5.  One  side  is  8  units 
long  and  at  an  angle  of  60° 
with  HS. 


Dra'w  the  prs.  of  a  regular 
octagon  on  plane  T.  Its  cen- 
ter is  point  d,  -8,  -5.  Its 
short  diameter  is  8  units  long 
and  parallel  to  H. 


'20 

Dra-w  the  prs.  of  a  circle 
tangent  to  lines  A  and  B.  Its 
diameter  is  8  units. 


Draw  the  prs.  of  a  circle 
tangent  to  lines  A  and  B.  Its 
diameter  is  lO  units. 


Draw  the  prs.  of  a  circle 
tangent  to  the  traces  of  plane 
M.    Its  diameter  is  1 2  units. 


Draw  the  prs.  of  a  circle 
tangent  to  the  traces  of  plane 
W,  and  lying  in  3Q.  Its  di- 
ameter is  16  units. 


Dra^  the  projections  of  the  figures  indicated.    (Art.  45,  page  35J.) 


Unit  of  measure,   }  inch.    Space  required  for  eacli  problem,   2J  x  3   inches.    Angles   between    GL   and   traces  of      Dl    ATC     A 
planes,  multiples  of  15°.    Measurements  from  GL,  in  light  type,   and  from  right-hand  division  line,   in  heavy  type.      "^-A  I   C.     O 


Plane  S  is  perpendicular  to  V 


Determine  the  prs.  of  the  line  of  intersection  between  planes  N  and  S.    (Arts.  49,  SO,  page  35,  Art.  52,  page  37.) 


Unit  of  measure,  }  inch.     Space  required -for  each  problem,  2J  x  3  inches.     Angles  betw^een   OL   and   traces   of      PI   ATE    7 
lianes,  multiples  of   15^.      Meaaurenaenta  from   GL,   In  light  type,   and  from  right-hand  divlaion  line,  In  heavy  type. 


HM 

H8 

VN 

VN 


9        8 


VS 


vs 


HH 


HS 


S  passes  through 
IQ  and  3QatC0°with  V 


S  passes  through 

2Q  and  4Qat30' 

with  H 


Determine  the  prs.  of  the  line  of  intersection  ber«-een  planes  N  and  S.    (Arta.  53-S6.  pages  37-40.) 
Problems  1-4.       Solve  by  Case  2.     (Art.  51,  page  36.) 
Problems  5-12.     Solve  by  Case  3.     (Art.  54,  page  38.) 


Unit  of  measure,  J  Inch.    Space   required    for   each  problem,   2.^x3   inches.     Angles   between    GL   and   traces   of      qi    a^c     O 
planes,  multiples  of  15°.    Measurements  from  GL,  In  light  type,  and  from  right-hand  division  line,  in  heavy  type.      "LA  I   C.    O 


^^1 

14;    loj 

6 

9^ 

HN 

\<^^ 

Il8 

^ 

2 
8 

\^       VN 

\<^«. 

10 


VN 


HN 


9        12 

-6 


22 


16 


Determine  the  pra.  of  the  point  in  which  line  A  pierces  plane   N.     Indicate   the    prs.   of   the  point  by  c^  and   c*^.      (Arts.    57-59, 
page  42.) 


Unit  of  zneasnre,  |  inch.    Space   required   for   each  problem,  2^  x  3  inches.     Angles  between  GIj  and  traces   of     Dl    ATP     Q 
I )lape8,  multiples  of  15-.    Meaaurementa  from  GL,  in  light  type,  and  from  right-hand  diviaion  line,  in  heavy  type.      '    ^'^  I  t    l7 


a -3 


VN 

6' '2 

!l2 

6*1 2 

1       HM 

—  5 

O  ^8 


b'l^ 


VN 


HM 


i|2 


a  -2 


b'U 


a'-r9 


bi-9 


6^8 


r* 


12 


a  -2 
a*  4 

6*6 

b'ls 


HM 


6t2 


a^8 


1 

-^ 

8    1 

61- — 
1 

<, 

J^> 

.6  ! 

'i^ 

-^^ 

6,  ;« 

^k 

0' 

1  1 
I3 1 

3|  ^^ 

1 
1 

^■^ 

1  1 
1  1 

8' 

B» 

^e! 

Problems  1-8.       Determine 
Problems  9-12.    Determine 


the  prs. 
the  prs. 


of  the  point  in  -which  line  ab  pierces  plane  N.    (Art.  60.  page  42.) 

of  the  point  In  which  line  A  pierces  the  plane  of  lines  B  and  D.    (Art.  61,  page  44.) 


Unit  of  measure,  |  Inch.     Space  required  for  each  problem,  2^x3   Inches.      Measurements   from   QL,   in   light      PIAfP     1A 
type,  and  from  right-hand  division  line,,  in  heavy  type. 


t^^-^^ 


io[\ 

\              B 

8  \ 

2      ^^    C 

191 

18           112 

6 

1 

2          ^1      B 

6' ' 

6 

>i^ 

1 

1       1 
2^    120 

14  !l2/ 

/^,2    1 

1      14 

1  k 

6/0 

10 
6 

2 
17 

1^ 

7 

6 
^0 

K            ' 

6| 

Q 

\ 

v|<, 

9  II 


Problems  1-4.       Determine  the  prs.  of  the  point  in  •which  the  line  pierces  the  polygon. 
Problems  5-15.    Determine  the  prs.  of  the  points  in  which  the  line  pierces  the  object. 


(Art.  61,  page  44.) 
(Art.  61,  page  44.) 


Unit  of  measure,  |  Inch.     Space  required  for  each  problem,  2^  x  3  inches.     Angles  between  OL  and  traces  of     Dl    A^C     11 
planes,  multiples  of   15-.     Measurements  from   GL,  in  light  type,  and  from  right-hand  division  line,  in  heav>' type.      •    LM  I  &      II 


20^ 

\ 

aj6       >^y^ 
y^ 

J 

^^ 

2 

afs 

'l4 

5 

6 

3 

17     ^\lO 

4- 

1 
1 

a''8 

^ 

J^y^ 

1                    "-^^ 
1                         ^ 

fl*-4 

1 

.' 

a-8 

^^ 

VH 

-8      5 

HN 

8      6 

h 

ar8 

y" 

.    7 

I6X    12 

8 

a'j6 

1          Hit 

k 

a-r6 

1 
1 

1 

!l2 

1 
1 
'12 

*1 

a-8 

1 

1        ^* 

i/l2 

5 

1 

0x8 

1 

.! 
a-8 

12              ^ 

9 

11/ 

/" 

ws 

-8" 

//S 

12 

1 

b\2 

f^ 

A 

12 

-4 

12 

Problems  1-8.  Determine  the  prs.  and  true  length  of  the  line,  ab,  measuring  the  shortest  distance  from  point  a  to  plane  N. 
(Arts.  62-65,  pages  44,  45.) 

Problems  9-12.  Determine  the  prs.  of  a  perpendicular,  ab,  to  plane  S  at  point  b  on  S.  Line  ab  to  be  8  units  long.  (Art.  36,  page 
24:  Arts.  62-65,  pages  44,  45.) 


Unit  of  measure,  \  inch.     Space   required   for  eacb    problem,  5 
type,  and  from  right-band  division  line,  in  heavy  type. 


7   inches.      Measurements   from   GL,   in   light       PIATF     10 


K 


-28- 


/'^^^ 


'4-v"r 


TV'I 


Problems  1,  2,  5-7.    Determine  the  prs.  of  the  shado-ws  of  the  object  on  itself  and  on  V  and  H.     (Arts.  66-69,  pages  46-48.) 
Problems  3,  4.    Determine  the  prs.  of  the  shadow  of  the  chimney  on  itself  and  on  the  roof. 
Problem  8.    Determine  the  prs.  of  the  shadoTV  of  the  bracket  on  itself  and  on  V. 


planes 


Unit  of  measure,  |  inch.    Space  required  for  each  problem,  2V  x  3  inches.       Angles   between  OL  and   traces   of      Dl    ATCT     1  *) 
nes,  multiples  of  15=.    Measurements  from  GL,  in  light  type,  and  from  right-hand  division  line,  in  heavy  type.      •    LM  I   C      I  O 


\ 

\                  «*6 

1 

\^ 

a'rl 

1 
1 

1 
1 

2 

-9     3 

-6 

HN 

4 

\       a +3 

a'j3 

\      !i« 

- 

\ 

i|2 

a.\ 

I6\^ 

16 

\i 

.12 

il2 

^\^ 

1^ 

a^6     \^ 

.•u 

VH 

9 

aVe 

5 

6 

V 

7 

<.'t»    ^ 

1         ? 

f^a'-A 

Xc'-* 

a'|2 

8 

- 

16 

/          1 
/                '8 

20!   16 

20 

8     16 

14          / 

7 

2 

.•U 

'K 

>< 

/     - 

9^ 

C" 

6't6 

1 
3  j 
6 

9 

5  • 

il8 

10 

1"' 

-9^ 

« 

II 

6' 

le 

7 
2 

-1 

12 

|I6            8 

..i  A. 

1                    '     1 

5\ 

;i2       6 

\l9 

1    d' 

1 

O 
6 

12 

-3 

8 

Problems  1-6.       Determine  the  traces  of  the  plane  containing  point  a  and  parallel  to  plane 
Problems  7-12.    Determine  the  traces  of  the  plane  containing  point  b  and  perpendicular  to 


N.     (Art.  71,  page  50.) 
line  C.    (Art.  72,  page  50.) 


Unit  of  measure,  |  inch.     Space  required  for  each  problem,  2|  x  3  inches.    Angles  between  GL  and  traces  of      PIATE     14 
planes,  multlplea  of   15°.     Measurements  from  GL,  in  light  type,  and  from  right-hand  division  line.ln  heavy  type. 

2 


9^8 


9 

\ 

\ 

%j^ 

^5 

18 

'",<'■ 

|8 

L- 

J 

6 

6 

8. 

1  \ 
1 
1  / 

^l 

.Y 

^2 

61    \'=2 

18 

12 

1      \    1 

12 


'r\ 

<\ 

[l8 

n 

\8 

I0| 

\ 

2*^^^ 

\^   1 

^ 

Problems  1-4.       Through  point  b  pass  a  plane  parallel  to  lines  A  and  B.    (Art.  73,  page  51.) 
Problems  5-8.       Through  line  A  pass  a  plane  parallel  to  line  B.     (Art  74,  page  52.) 
Problems  9-12.    Through  line  A  pass  a  plane  perpendicular  to  plane  N.    (Art.  75,  page  62.) 


Unit  of  measure,  J  inch.    Space   required   for   each   problem,   6x7   inches.     Measurements   from   GL,   in  light     qi    atC     1R 
le.  and  from  risrht-hand  division  line,  in  heavv  tvne.  ~  imf\  I   C>      t  \f 


type,  and  from  right-band  division  line,  in  heavy  type 


fiil4 


4 


1 

■           J^ 

19 

\^'P 

27 

I8\|4 

8 

\^«.;    ; 

1^8   J 
1     «  / 

I      7 

6 

2* 


a  •'-16 


Determine  the  prs.  and  true  length  of  the  line  measuring  the  shortest  distance  bet'sveen  lines  A  and  B.    (Art.  79,  page  54.) 
Note.  —  The  problenti  is  best  solved  by  passing  the  auxiliary  plane  through  line  A, 


Unit  of  measure,  }  Inch.    Space  required  for  each  problem,  2^x3   inches.    Angrles   bet-ween   GL   and   traces  of     qi    atc    1  A 
planes,  multiples  of  15°.    Measurements  from  GL,  in  light  type,  and  from  right-hand  division  line,  in  heavy  type.      •    ^'^  •   C.     I  O 


1 

/ 

2 

3 

9n 

/4 

2oX     ^^                   14 

20/     ||6 

-4i 

1 
!6 

- 

I6|\ 

J4                6 

|22 

iVii 

1 

1 

1 
1 
1 

1     ^ 

e^"""^ 

1 

7 

s'^           ^^ 

^\ 

81^ 

^^8 

^ 

6 

8r\                HN                   ^ 

HN 

/!% 

6 

f 

6^ 

/9                        7 

afs 

V\^ 

8 

1                        ^^^^v^ 

^ 

j20                          ^^1 

> 

16 

- 

110/ 

a- 

a" 

2^ 

^ 

|4 

1 

1 

1         ^ 

^ 

< 

12 

■3 

^ 

12 

4 

■6 

1^ 

VN        ^^^'S 

a" 

/l9         9 

•71-^.^^ 

>^° 

10 

=N 

li 

a" 

9 

12 

je               !6 

K 

^;. 

- 

1 
1 

.            \     6 

i^ 

U 

1 
2L                 1 

16 

6 

18 

^\  N2 

a\2 

^■^^^      1 

^^4 

a 

^"^6 

6"^ 

7 

Problems  1-8.     Determine  the  true  size  of  the  angle  between  line  A  and  plane  N.    (Art.  80,  page  54.) 

Problems  9-12.    Determine  the  true  size  of  the  angle  bet-^reen  line  B  and  V  and  H.     (Art.  81,  page  55.)     Note.  —  Iietter  the  angle 
with  V  as  X,  and  -with  H  as  y. 


Unit  of  measure,   }  Inch.      Space  reqviired.  for  each  problem,  5x7  inches.     Angles  between  GL  and  traces  of 
planes,  multiples  of  15^.    Measurements  from  GL,  in  light  type,  and  from  right-hand  division  line,  in  heavy  type. 


PLATE   17 


33  28 


13  8 


Problems   1-4.    Determine  the  tme  size  of  the  diedral  angle  between  planes  M  and  T.    (Arts.  83-85  pages,  57,  58.) 
Problems  5-8.    Determine  the  true  sizes  of  the  diedral  angles  of  the  objects.    (Arts.  83-35,  pages  57,  58.) 


Unit  of  measure,  J  inch.    Space   required   for   each  problem,   2.^  x  3  inches.    Angles   between  OL  and  traces  of      pi    AXP     1ft 
planes,  multiples  of  15°.    Measurements  from  GL,  in  light  type,  and  from  right-hand  division  line,  in  heavy  type.      r  L_M  It      1  O 

\l4 

2 

4 

^^->^ 

\^ 

Jp^ 

7 

5 

/ 
12/ 

6 

% 

7 

8 

7 

16/ 

7 

^y^ 

^y^ 

2 

9 

V8                  ,   10 

II 

ys               3    12 

"^                   3 

"^               n 

^8                  , 

1 

Determine  the  angle  between  plane  S  and  V  and  H..  (Art.  86,  page  68.)    Note.  — Letter  the  angle  with  V  as  x,  and  -with  H  as  y. 


Unit  of  measure,  I  inch.    Space  required  for  the  problem,  7  x  lO  inches. 


PLATE  19 


HIP   RAFTER. 

Determine     length  —  down     cut- 
heel  cut —  side  cut  —  top  bevel. 

JACK    RAFTER. 

Determine  side  cut. 

PURLIN. 

Determine  down  cut  —  tide  cut- 
angle  between  face  and  end. 


Determine  the  bevels,  cuts,  and  lengths  of  roof  members,  aa  above.    (Art.  87,  page  60.) 


Unit  of  measure,  }  inch.    Space  required  for  the  problem,  7  x  lO  inches. 


PLATE  20 


>./^. 


R.  —  Plane  of  web  of  Hip  Rafter. 
P.  ^  Plane  of  web  of  Purlin. 
S.  —  Plane  perpe/idicular  to  line  of  in- 
tersection between  R  and  P. 


Determine  angrle  of  cut  on  top  of  purlin  (A).    Bevel  on  -web  of  purlin  (B).    Angle  befween  plane  of  -web  of  hip  rafter  and  purlin,  or 
bend  of  gusset  (C).    Angle  between  top  edges  of  gusset  (D).    (Art.  87,  page  61.) 


Unit  of  measure,   j  inch.      Space  required  for  each  problem,   2i  x  3  inches.     Angles  between  GL  and  traces  of      Ol    ATC     01 
planes,  multiples  of  15~".     Measurements  from  GL,  in  light  type,   and  from  right-hand  division  line,  in  heavy  type.       •    L.r\  I   ^     A  I 


" 

^       2 

60°with  H.                                      ^ 

TS'with  H.                                       * 

\ 

45°with  H. 

30°with  V. 

HH 

7 

5 

/                    6 

/                            ^ 

7 

8 

^                         60°with  H. 

60°vyith  H. 

eo'with  V. 

30°with  V. 

9 

10 

II 

12 

eo'with  V. 

45'with  H. 

75°w;th  H. 
45°with  V. 

30°v»ith  H. 

eo'with  V. 

90°with  H. 
45'with  V. 

Problems  1-8. 
Problems  9-12. 


Determine  one  position  of  the  missing  trace  of  plane  N.    (Arts.  88,  89,  page  62.) 

Determine  the  traces  of  plane  S  making  the  g^iven  angles  t^tb  V  and  H.    (Art.  90,  page  62.) 


Unit  of  measure,  J  Inch.    Space  required  for  each  problem,  7  X  lO  inches. 


PLATE  22 


In  belt  transmission  one  condition  must  always  be  obtained,  namely  :  the  point  on  a 
pulley  from  which  the  belt  is  delivered  musr  lie  in  the  mid-plane  of  the  pulley  to  which 
it  is  delivered.  If  the  shafts  are  at  right  angles  and  placed  as  in  the  figure,  they  will  run 
in  the  direction  indicated,  point  b  lying  in  the  mid-plane  of  the  large  pulley,  and  point 
c  lying  iri  the  mid  plane  of  the  small  pulley  But  if  the  direction  be  reversed  a  guide 
pulley  would  be  necessary  to  compel  the  belt  to  fulfill  the  above  conditions.  The  point 
at  which  the  direction  of  the  belt  Is  changed  by  the  guide  pulley  is  governed  by  con- 
venience. Problem  I  is  that  of  a  steering  gear  and  requires  one  guide  pulley.  Prob- 
lem 2  illustrates  a  condition  which  necessitates  two  guide  pulleys,  although  but  one  is 
to  be  determined. 


Problem  1.  Draiv  the  projections  of  a  guide  pulley  8  units  diameter,  2  units  face,  and  determine  the  shaft  angle  with  V  and  H. 
(Art.  45,  page  32.) 

Problem  2.  Dra-w  the  projections  of  one  guide  pulley  8  units  diameter,  2  units  face,  and  determine  the  shaft  angle  with  V  and  H. 
(Art.  46,  page  32.) 


Unit  of  measure,  |  inch.    Space  required  for  each  problem,  7  x  lO  Inches. 


PLATE  23 


Locate  GL  38  units  from  lower  margin  line.  Locate  point  b  on  GL 
16  units  from  right-hand  margin  line. 

Draw  the  prs.  of  a  regular  hexagonal  prism  in  IQ,  resting  on  a  plane, 
the  vertical  trace  of  which  makes  an  angle  of  IS-"  with  GL,  and  the 
horizontal  trace,  an  angle  of  45^  with  GL.  The  center  of  the  base  of 
the  prism  is  a  point  of  the  plane,  17,  4  The  sides  of  the  hexagon  are  6 
units  and  two  of  the  sides  are  perpendicular  to  the  horizontal  trace  of 
the  plane  Altitude  of  prism  is  12  units.  The  traces  of  the  plane 
intersect  GL  in  point  b. 

Determine  the  shadow  of  the  pnsm  on  the  plane. 


Locate  GL  38  units  above  lower  margin  line.  Locate  point  b  on  GL 
20  units  from  righi-hand  margin  line. 

Draw  the  prs.  of  a  regular  pentagonal  pyramid  in  IQ,  resting  on  a 
plane,  the  vertical  trace  of  which  makes  an  angle  of  15°  with  GL,  and 
the  horizontal  trace,  an  angle  of  45°  with  GL  The  pyramid  rests  on 
its  apex  at  a  point  17,  4  Axis  of  pyramid  is  perpendicular  to  the  plane 
and  is  12  units  long.  Circumscribing  circle  of  base  of  pyramid  is  12 
units  in  diameter.  One  side  pf  pentagon  to  be  parallel  to  H.  The 
traces  of  the  plane  intersect  GL  in  point  b. 

Determine  the  shadow  of  the  pyramid  on  the  plane. 


Locate  GL  38  units  from  lower  margin  line.  Locate  point  b  on  GL 
16  units  from  right-hand  margin  line. 

Draw  the  prs.  of  a  regular  hexagonal  pyramid  in  IQ,  resting  on  a 
plane  which  makes  an  angle  of  20'  with  H  and  75°  with  V.  The  pyra- 
mid rests  on  its  apex  at  a  point  in  the  plane,  18,  4.  Axis  of  pyramid  is 
perpendicular  to  the  p'ane,  and  is  12  units  long.  The  short  diameter  of 
the  base  is  parallel  to  H;  the  long  diameter  is  12  units  in  length.  The 
traces  of  the  plane  intersect  GL  in  point  b. 

Determine  the  shadow  of  the  pyramid  on  the  plane. 


Locate  GL  38  units  from  lower  margin  line.  Locate  point  b  on  GL 
60  units  from  right-hand  margin  line. 

Draw  the  prs.  of  a  cube  in  IQ,  resting  on  a  plane  which  makes  an 
angle  of  SO''  with  H  and  65°  with  V.  The  center  of  the  base  of  the 
cube  is  10  units  from  the  horizontal  trace  of  the  plane  and  8  units  from 
V.  One  diagonal  of  the  base  is  parallel  to  H.  Edge  of  cube  is  8  units. 
The  traces  of  the  plane  intersect  GL  in  point  b. 

Determine  the  shadow  of  the  cube  on  the  plane. 


Dra'w  the  prs.  of  a  solid  resting  on  an  oblique  plane,  and  determine  the  prs.  of  its  shadow  on  the  plane. 


Unit  of  measure,  |  inch.    Space  required  for  each  problem,  5x7  Inches.    Measurements  from  GL,  in  light  type,      pi    AfC     OA 
and  from  right-hand  division  line,  in  heavy  type. 


Problems  1,  2,  8.    Draw  the  traces  of  a  plane  which  is  tangent  at  point  a  of  the  surface.     (Arts.  104,  105,  pages  72-74.) 

Problems  3,  4,  7.     Dra-w  traces  of  planes  w^hich  are  tangent  to  the  surface  and  contain  point  a.     (Art.  106,  page  74  ;  Art.  108,page75.) 

Problems  5,  6.    Draw^  traces  of  planes  which  are  tangent  to  the  surface,  and  parallel  to  line  B.     (Art.  107,  page  75  ;  Art.  109,  page  76.) 


Unit 
and  from 


of  me&sure,  |  inch.    Space  required  for  e»cii  problem,  5x7  inches.    Measarements  from  Gt>.  in  light  typ«,      qi    A'^C     f^ti 
im  right-hand  division  line,  in  heavy  tvpe.  r  ^^^  I   &     Axl 


Problems  1-4.    Dra^e  the  traces  of  a  plane  •which  is  tangent  at  point  a  of  the  snrface.    (Art.  Ill,  page  78.) 

Problems  5,  6.    Draw  traces  of  planes  containing  point  a  and  tangent  to  the  surface  at  given  parallel.     (Art.  112,  page  79.) 

Problems  7,  8.    Dra-n-  the  traces  of  planes  tangent  to  the  sphere  and  containing  line  A.    (Art.  114    page  80.) 

Note.    Problems  5-8.    Determine  the  points  of  t»ngency. 


Unit  of  measure,  J  inch.     Space  required  for  each  problem,  7  x  10  inches.     Angles  bet-ween  GL  and  traces  of 
planes,  multiples  of  15°.    Measurements  from  GL,  In  light  type,   and  from  right-hand  division  line,  In  heavy  type 


PLATE  26 


Determine  the  Intersection  of  the  plane  -with  the  solid,  and  develop  the  surface. 

Problem  1.     (Arts.  120,  121,  pages  83,  84.)       Problem  2.     (Art.  127,  page  92.)       Problem  3.     (Arts.  124,  125,  page  88.)       Problem  4. 
(Arts.   122,   123,  page  86.) 


Unit  at  measure,  }  inch.    Space  required  for  eacb  problem,  7  x  lO  inclies. 


PLATE  27 


The  generatrix    makes    an    angle 
of  15^  with  H,  and  one  revolution. 


-31- 


Pitch  12  units ;  length  of  longest 
elefnent,  '9  units. 


Problems  1,  2.    Dra-tr  and  develop  an  taelical  convolute.    (Arts.  129-131,  pa^es  93-05.) 

Problems  3,  4.    Delvelop  one  quarter  of  the  lamp  shade.     Problem  3.    (Art.  126,  page  91.)     Problem  4.    (Art.  123,  page  86.) 


Unit  of  measure,   J  inch.     Space  required  for  each  problem,   5x7  inches.     Angles  bet\veen  GL  and  traces  of     qi    a  "re     Oft 
planes,  multiples  of  15°.    Measurements  from  GL,  in  light  type,  and  from  right-hand  division  line,  in  heavy  type,      i    1_M  I   H     ^O 


A  is  the  generatrix  of 
an  hyperboioid  of  revolu- 
tion. 


Determine  the  intersection  of  the  plane  N  with  the  double-curved  surface  of  revolution.    (Art.  132,  page  96.) 


Unit  of  measare,  J  inch.    Space  reqtdred  for  each  problem,  7  x  lO  inches.    Measnrementa  from  OL,  in  U^ht  type,     qi    A"T"P     OQ 
and  from  right-hand  division  line,  in  heavy  type.  r  l-^  I   C    ^7 


Determine  the  intersection  between  the  solids. 


Unit  of  measure,  J  Inch.    Space  required  for  each  problem,  7  x  lO  inches.    Measurements  from  GL,  in  light  type, 
and  from  right-hand  division  line,  in  heavy  type. 


PLATE  30 


37  28      16 


35   28 


Determine  the  intersection  betiveeu  the  solids. 


t— -22— , 


Unit  of  measure,  J  inch.    Space  required  for  eacb  problem,  6x7  inches.    Measurement3  from  GL,  in  light  type, 
and  from  right-hand  division  line,  in  heavy  type. 


PLATE  31 


29  1 

4— «r       7 

- 1 

^ 

'IS 

T 

1 

V 

V 

1^ 

in 

Determine  the  intersection  bet^veen  the  solids. 


The  unit  of  measurement  is  1/16  inob.    Space  required  7  x  lO  inches. 


PLATE  32 


Oeveloprnent  of 

I 

Dome  drawn  here. 


A  sectional  and  end  view  of  a  portion  of  a  boiler  is  shown.  It  is  reqi/ired  to  develop  one  half  of  the 
dome,  and  that  part  of  the  slope  sheet  lying  between  the  elements  ef  and  cd,  adding  the  amount  necessary 
for  flanging  or  lap 

_  Make  two  full  views  of  the  boiler,  indicating  the  thickness  of  plate  by  a  single  line.  Determine  the 
projection  of  the  two  curves  of  the  flange  ab,  observing  that  the  radial  elements  ot  the  flange  make  angles 
varying  from  0^  to  y^.  The  elements  of  the  slope  sheet  will  be  parallel  to  each  other  and  to  the  vertical 
plane,  thus  appearing  in  their  true  lengths  in  vertical  projection.  The  development  may  be  obtained  by 
either  of  the  followingf  methods,  the  first  being  that  used  in  practice.  First  method:  the  elements  being 
parallel  to  V,  the  development  may  be  obtained  .as  in  Art.  126,  page  91,  the  amount  indicated  for  the  lap 
being  added  to  the  sheet.  Second  method  :  obtain  a  right  section  VX  by  Art  118,  page  82,  the  development 
will  be  a  right  line,  mn.  To  obtain  the  distance  between  the  elements  revolve  the  plane  of  the  right  section 
to  P,  and  measure  the  required  distances  in  the  profile  projection,  or  end  view.     Develop  as  in  Art.  125,  page  8S 


Draw  and  develop  the  dome  and  connection  sheet,  or  slope  sheet,  of  a  locomotive  boiler. 


Unit  of  measure,   |  Inch.    Space  required  for  each   problem,  5x7  inches.     Angles  between  GL  and  traces  of       pi    A-rC     QO 
planes,  multiples  of  15^.    Measurements  from  GL,  in  light  type,  and  from  right-hand  division  line,  in  heavy  type.       ~  *~'^  '  ^    '**' 


Problems  1,  2.  I>Taw  an  element  of  fhe  -warped  surface  through  point  a.    (Art.  145,  page  llO.) 

Problems  3,  4.  Draw  an  element  of  the  warped  surface  through  point  a.    (Art.  146,  page  111.) 

Problems  5,  6.  Draw  an  element  of  the  warped  surface  parallel  to  line  C  of  the  plane  director,  N.    (Art.  147,  page  112.) 

Problems  7,  8.  Draw  an  element  of  the  hyperbolic  paraboloid  through  point  a,  of  a  directrix.    (Art.  ISO,  page  116.) 


Unit  of  measure,  |  Inch.     Space  required  for  each  problem,  6x7  Inches.     Angles  between  GL  and   traces   of     ni    ATP     "^A 
planes,  multiples  of  16°.    Measurements  from  GL,  in  light  type,  and  from  right-hand  dlvison  line,  in  heavy  type.      '    ^"  '  ^    "^ 


A  is  the  generatrix  of  an  hyper- 
boloid  of  revolution. 


A  Is  the  generatrix  of  an  hyper- 
boloid  of  revolution. 


A  is  the  generatrix  of  an  hyper- 
boloid  of  revolution. 


A  is  the  generatrix  of  an  hyper- 
boloid  of  revolution. 


/        12 

Z-. 

1 

y 

12 

3e~- 

to 

/ 

10 

22 

C  "^ 

\ 

A 

\^c 

M 

nor  Axis 

12 

X 

Problems  1,  2.  Draw  an  element  of  the  hyperbolic  paraboloid  through  point  a.    (Art.  161,  page  116.) 

Problems  3,  4.  Draw^  an  element  through  point  a.     (Art.  156,  page  120.) 

Problems  6,  6.  Draw  a  plane  tangent  to  the  surface  at  point  a.    (Art.  159,  page  121.) 

Problems  7,  S.  Througli  line  A  pass  a  plane  tangent  to  the  surfaca.    (Art.  160,  page  122.) 


.»i^. 


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